Questions on Word Problems: Geometry answered by real tutors!

Tutors Answer Your Questions about Geometry Word Problems (FREE)

Question 945236: find the measures of the angles of a triangle if the measure of one side is twice the measure of a second and the third angle measures two times the second decreased by 20
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13367)   (Show Source):
You can put this solution on YOUR website!

find the measures of the angles of a triangle if the measure of one angle is twice the measure of a second and the third angle measures two times the second decreased by 20

One angle (the "first" angle) has a measure equal to twice the measure of the second angle; the third angle has a measure that is 20 less than twice the measure of the second angle. So

x = second angle measure
2x = first angle measure
2x-20 = third angle measure

The sum of the angles of a triangle is 180 degrees:

(x)+(2x)+(2x-20) = 180
5x-20 = 180
5x = 200
x = 40

ANSWERS: the angle measures in degrees are x=40, 2x=80, and 2x-20 = 60

Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
find the measures of the angles of a triangle if the measure of one side is twice
the measure of a second and the third angle measures two times the second decreased by 20
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The problem's formulation is incorrect.

If to read it word-in-word as it is written, it is a compote of words, but not a Math problem.

My advise to the managers of this project is to remove this Math composer
from writing Math problems IMMEDIATELY.

It should be done 7 years ago - next day as you hired him.

Question 1159001: Recall that the lengths of the sides of triangle ABC are often abbreviated by writing a = BC, b = CA, and c = AB. Sketch triangle ABC where angle BCA is right and mark F as the foot of the perpendicular drawn from C to the hypotenuse AB. In terms of a, b, and c, express the lengths of FA, FB, and FC. The equation c = FA+FB can be used to check your work.
Answer by KMST(5396)   (Show Source):
You can put this solution on YOUR website!
This is our first triangle: triangle ABC
When I draw the perpendicular from C to hypotenuse AB, we will have point F, and the first triangle will be split into 2 triangles.
I will labeled them as triangle #2 and triangle #3.

Triangle ABC, triangle #2, and triangle #3 are similar triangles. They have the same shape, the same angle measures, and the same ratios of corresponding sides.
For the ratio of side opposite to hypotenuse we have:
-->
For the ratio of side opposite to hypotenuse we have:
-->
Of course, we know that ,
We can verify that the expressions we found for and are correct, by substituting, and finding that it agrees with what we know:
From --> --> --> .
However, along the way, we find that we proved the Pythagorean theorem from our knowledge about similar triangles.
We can also prove that <--> .

Question 1163329: If you are in shallow water and look at an object below the surface of the water,
the object will look farther away from you than it really is. This is because as light passes
through air and water it bends or refracts the light rays. The index of refraction for water is 1.33.
This is the ratio of the sine of theta 1 and the sine of theta 2.

a)You are in 2 feet of water in the shallow end of a pool. You look down at some goggles at angle
(theta=60) degrees measured from a line perpendicular to the surface of the water. Find (theta 2 )

b) Find the distances x and y

Found 3 solutions by KMST, ikleyn, CPhill:
Answer by KMST(5396)   (Show Source):
You can put this solution on YOUR website!
I agree with ikleyn and CPhill, but I would like to draw and overexplain as usual.

(given)
value calculated based on Snell's Law:
-->-->
From that we can find (rounded)
We can also find ,
, and the distance
(rounded)
Independently of refraction, we can find the distance
(rounded)

Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
If you are in shallow water and look at an object below the surface of the water,
the object will look farther away from you than it really is. This is because as light passes
through air and water it bends or refracts the light rays. The index of refraction for water is 1.33.
This is the ratio of the sine of theta 1 and the sine of theta 2.

a) You are in 2 feet of water in the shallow end of a pool. You look down at some goggles at angle
(theta=60) degrees measured from a line perpendicular to the surface of the water. Find (theta 2 )

b) Find the distances x and y
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

As the incoming post is worded, it is mathematically incorrect,
because is asks to find distances 'x' and 'y',
but does not determine what these distances are.

It is why this posted problem recalls me a lame horse with three legs.

Answer by CPhill(2264)   (Show Source):
You can put this solution on YOUR website!
```python?code_reference&code_event_index=2
import math
# Given values
n_air = 1.0
n_water = 1.33
theta_1_deg = 60.0
depth_y = 2.0 # feet
# Part a: Find theta_2
# n1 * sin(theta1) = n2 * sin(theta2)
theta_1_rad = math.radians(theta_1_deg)
sin_theta_2 = (n_air * math.sin(theta_1_rad)) / n_water
theta_2_rad = math.asin(sin_theta_2)
theta_2_deg = math.degrees(theta_2_rad)
# Part b: Find distances x and y
# y is the actual depth
y = depth_y
# x is the horizontal distance from the entry point to the object
# From the right triangle in the water: tan(theta_2) = x / y
x = y * math.tan(theta_2_rad)
print(f"{theta_1_deg=}")
print(f"{theta_2_deg=}")
print(f"{y=}")
print(f"{x=}")

```
```text?code_stdout&code_event_index=2
theta_1_deg=60.0
theta_2_deg=40.62813065148206
y=2.0
x=1.715911347328087
```
To solve for the angle of refraction and the geometric distances, we use **Snell's Law**, which describes how light bends when crossing the boundary between two different media (in this case, air and water).
### (a) Find $\theta_2$
Snell's Law is given by the formula:
$$n_1 \sin \theta_1 = n_2 \sin \theta_2$$
Where:
* $n_1$ is the index of refraction for air ($\approx 1.00$).
* $n_2$ is the index of refraction for water ($1.33$).
* $\theta_1$ is the angle of incidence in air ($60^\circ$).
* $\theta_2$ is the angle of refraction in water.
Using the ratio provided in the problem ($1.33 = \frac{\sin \theta_1}{\sin \theta_2}$):
$$1.33 = \frac{\sin 60^\circ}{\sin \theta_2}$$
$$\sin \theta_2 = \frac{\sin 60^\circ}{1.33}$$
$$\sin \theta_2 = \frac{0.8660}{1.33} \approx 0.6511$$
$$\theta_2 = \arcsin(0.6511) \approx 40.63^\circ$$
**The angle of refraction $\theta_2$ is approximately $40.63^\circ$.**
---
### (b) Find the distances $x$ and $y$
In this geometric model of refraction:
* **$y$** represents the vertical depth of the water. According to the problem, you are in 2 feet of water.
$$y = 2\text{ ft}$$
* **$x$** represents the horizontal distance from the point where the light ray enters the water to the location of the goggles on the pool floor.
Using the right triangle formed by the light ray under the water:
$$\tan \theta_2 = \frac{\text{opposite}}{\text{adjacent}} = \frac{x}{y}$$
$$x = y \cdot \tan \theta_2$$
$$x = 2 \cdot \tan(40.63^\circ)$$
$$x \approx 2 \cdot 0.8580$$
$$x \approx 1.716\text{ ft}$$
**Final Results:**
* **$\theta_2 \approx 40.63^\circ$**
* **$y = 2\text{ ft}$** (Actual depth)
* **$x \approx 1.72\text{ ft}$** (Horizontal distance)

Question 1157992: The reflection property of parabolas. Consider the parabola whose focus is F = (1,4) and whose directrix is the line x = −3.
(a) Sketch the parabola, and make calculations that confirm that P = (7, 12) is on it.
(b) Find the slope of the line μ through P that is tangent to the parabola.
(c) Calculate the size of the angle that μ makes with the line y = 12.
(d) Calculate the size of the angle that μ makes with segment F P .
Answer by KMST(5396)   (Show Source):
You can put this solution on YOUR website!
(a) Sketch the parabola, and make calculations that confirm that P = (7, 12) is on it.

A parabola is the locus of the points that are at the same distance from the directrix and the focus.
The distance between directrix and is

The distance between focus and is

is at the same distance from the directrix and the focus,
so it is on the parabola.
We could prove P is on the parabola from the equation of the parabola.
Knowing that the equation of a parabola with its vertex at the origin and focal distance is
for parabolas with the x-axis as an axis of symmetry,
we can translate that equation for a parabola with the vertex at (-1,4), halfway between focus and directrix.
Doing that, we found that our parabola has and the equation for our parabola would be
-->
For we get -->

(b) Find the slope of the line μ through P that is tangent to the parabola.
We can estimate the slope of the tangent line from the graph or calculate it from the derivative of the function.
From --> we can find the derivative
.
For , that derivative is
For the function graphed in red above, is the derivative and slope of the tangent at point P.

(c) Calculate the size of the angle that μ makes with the line y = 12.
(d) Calculate the size of the angle that μ makes with segment F P .

The line slope is zero, it is parallel to the x-axis.
Line with slope makes an angle with the x-axis and with the line such that
-->
Segment FP, connecting } and , and line FP, have a slope of
. A line, or segment with such a slope would make an angle with the x-axis and line such that
-->
The angle that μ makes with segment FP is

Question 1157988: Let WISH be a cyclic quadrilateral, and K be the intersection of its diagonals WS and HI. Given that arc WI is 100 degrees and arc SH is 80 degrees, find the sizes of as many angles in the figure as you can.
Answer by KMST(5396)   (Show Source):
You can put this solution on YOUR website!
Am I missing something?
I cannot figure out what that WISH quadrilateral looks like.
Like this: ? Or like this: ? Or ... ?

Question 992859: Two sides of a triangle are 3ft and 6ft longer then the third side. If the perimeter of the triangle is 36ft find the lenght of each side
Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
Two sides of a triangle are 3ft and 6ft longer then the third side.
If the perimeter of the triangle is 36ft find the lenght of each side
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        This problem is solved incorrectly in the post by @lwsshak3.
        For correct solution, see what follows in my post.

let x be the length of3rd side
x+3 = 2nd side
x+6 = 1st side

x + (3+x) + (x+6) = 36

3x + 9 = 36
3x = 27
x = 9
3rd side is 9 ft long.
1st side is 9+6 = 15 ft long
2nd side is 9+3 = 12 ft long.

For these side lengths, the triangle inequalities are valid, so such triangle does exist.

Solved correctly for your right teaching/learning.

Question 1163755: Let u = [2, −3, 1] and v = [0, 1, 4]. Calculate the vector u − v. Place u and v tail-to-tail to form two sides of a triangle. With regard to this triangle, what does u − v represent? Calculate the number u•u and discuss its relevance to the diagram you have drawn. Do the same for the number (u − v)• (u − v).
Answer by CPhill(2264)   (Show Source):
You can put this solution on YOUR website!
### 1. Calculation of $u - v$
To subtract vector $v$ from $u$, we subtract the corresponding components:
$$u - v = [2 - 0, \ -3 - 1, \ 1 - 4] = [2, \ -4, \ -3]$$
### 2. Geometric Representation
When $u$ and $v$ are placed **tail-to-tail**, they form two sides of a triangle.
* The vector $u - v$ represents the **third side** of that triangle.
* Specifically, it is the vector that points from the **head of $v$** to the **head of $u$**.
---
### 3. Analysis of $u \cdot u$
**Calculation:**
$$u \cdot u = (2)^2 + (-3)^2 + (1)^2 = 4 + 9 + 1 = 14$$
**Relevance to the Diagram:**
The dot product of a vector with itself is equal to the square of its magnitude ($u \cdot u = |u|^2$).
* In your triangle, $u \cdot u$ represents the **squared length of the side formed by vector $u$**.
* To find the actual length of that side of the triangle, you would take $\sqrt{14} \approx 3.74$.
---
### 4. Analysis of $(u - v) \cdot (u - v)$
**Calculation:**
Using our result for $u - v = [2, -4, -3]$:
$$(u - v) \cdot (u - v) = (2)^2 + (-4)^2 + (-3)^2 = 4 + 16 + 9 = 29$$
**Relevance to the Diagram:**
Similar to $u \cdot u$, this represents the **squared length of the third side of the triangle** (the side connecting the tips of $u$ and $v$).
* $|u - v|^2 = 29$.
* The length of this side is $\sqrt{29} \approx 5.39$.
---
### Summary Table
| Expression | Vector Result | Scalar Value (Dot Product) | Geometric Meaning |
| :--- | :--- | :--- | :--- |
| **$u$** | $[2, -3, 1]$ | $14$ | Squared length of side 1 |
| **$v$** | $[0, 1, 4]$ | $17$ | Squared length of side 2 |
| **$u - v$** | $[2, -4, -3]$ | $29$ | Squared length of side 3 |
> **Note:** If you apply the Law of Cosines to this triangle ($c^2 = a^2 + b^2 - 2ab \cos \theta$), you'll find that these dot products are exactly the $a^2, b^2,$ and $c^2$ terms!
Would you like to calculate the angle $\theta$ between $u$ and $v$ using these values?

Question 1163922: Let’s assume the following statements are true: Historically, 75% of the five-star football recruits in the nation go to universities in the three most competitive athletic conferences. Historically, five-star recruits get full football scholarships 93% of the time, regardless of which conference they go to. If this pattern holds true for this year’s recruiting class, answer the following:
a. Based on these numbers, what is the probability that a randomly selected five-star recruit who chooses one of the best three conferences will be offered a full football scholarship?
b. What are the odds a randomly selected five-star recruit will not select a university from one of the three best conferences? Explain.
c. Explain whether these are independent or dependent events. Are they Inclusive or exclusive? Explain.
Answer by CPhill(2264)   (Show Source):
You can put this solution on YOUR website!
Based on the statistics provided, here are the calculations and logical breakdowns for this year's recruiting class:
### a. Probability of a Scholarship in a Top Conference
The probability is **93%** (or **0.93**).
**Reasoning:** The prompt states that five-star recruits get full scholarships 93% of the time **"regardless of which conference they go to."** This implies that the scholarship rate and the conference choice are treated as independent factors in this dataset. Therefore, knowing the recruit chose a top conference does not change the established 93% probability.
---
### b. Odds of Not Selecting a Top Conference
The odds are **1 to 3** (or $1:3$).
**Explanation:**
1. **Probability ($P$):** We know 75% ($3/4$) of recruits choose a top conference. Therefore, the probability of **not** choosing one is 25% ($1/4$).
2. **Calculating Odds:** Odds are expressed as the ratio of *Successes* to *Failures* (or in this case, *Not Selecting* vs. *Selecting*).
* $\text{Odds} = \frac{P(\text{Not Top})}{P(\text{Top})} = \frac{25\%}{75\%} = \frac{1}{3}$
3. **Interpretation:** For every 1 recruit who chooses a school outside the top three conferences, there are 3 recruits who choose a school within them.
---
### c. Event Relationships
**Independent vs. Dependent**
These are **Independent** events.
* **Why?** The prompt explicitly states the scholarship rate is 93% "regardless of which conference they go to." This means the outcome of the first event (choosing a conference) has no effect on the probability of the second event (getting a scholarship).
**Inclusive vs. Exclusive**
These are **Inclusive** events.
* **Why?** Mutually exclusive events are things that cannot happen at the same time (like a coin landing on both Heads and Tails). In this scenario, a recruit **can** both choose a top conference **and** receive a full scholarship simultaneously. Since these two outcomes can overlap in a single person, they are inclusive.
***
Do you have a specific recruit's data you're trying to model, or are you looking for the combined probability of both events happening at once?

Question 300629: How Many 2 Inch Square Blocks Can Fit Inside A 10inch By 20 Inch Box? how do u get the answer?
Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
How Many 2 Inch Square Blocks Can Fit Inside A 10inch By 20 Inch Box? how do u get the answer?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        @mananth interprets this problem as 3D, although the problem does not give any reason for it.
        I will interpret and solve this problem as 2D for plane figures.

Place 5 blocks along the 10-inches dimension and 10 blocks along the 20-inches dimension. In total, you will get 5 x 10 = 50 blocks. ANSWER

Question 280034: From the top of a 210 ft lighthouse, the angle of depression to a ship in the ocean is 21 degrees. How far is the ship from the base of the lighthouse to the nearest tenth of a foot?
Found 2 solutions by MathTherapy, ikleyn:
Answer by MathTherapy(10858)   (Show Source):
You can put this solution on YOUR website!

From the top of a 210 ft lighthouse, the angle of depression to a ship in the ocean is 21 degrees. How far is the ship from the base of the lighthouse to the nearest tenth of a foot? ************************ This author is applalled, looking at some of these "people's" responses. Certain things make absolutely NO SENSE!! The smaller acute angle is , and is obviously opposite the SHORTER LEG. This makes the SHORTER LEG the HEIGHT of the lighthouse, at 210 feet. This means that the LONGER leg, which is opposite the LARGER ACUTE ANGLE, will be the DISTANCE from the ship to the base of the lighthouse. As that this is a FACT, how come the other "person" responds with an answer of 140 feet for the DISTANCE from the ship to the base of the lighthouse, when the length of this LEG should be GREATER than 210 feet? This makes absolutely NO SENSE. Do these "people" ever check or just use common sense to determine if their answers make sense? Obvioulsly NOT!!

Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
From the top of a 210 ft lighthouse, the angle of depression to a ship in the ocean is 21 degrees.
How far is the ship from the base of the lighthouse to the nearest tenth of a foot?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        The solution in the post by @mananth is incorrect.
        I came to bring a correct solution.

The angle of depression from the top of tower = angle of inclination from the ship
tan(21°) = 210/distance from foot of light house
distance = 210/tan(21°) = 210/0.38386403503 = 547 ft (rounded).         ANSWER

Solved correctly.

Question 272686: A boat is 1000 meters from a cliff. If the angle of depression from the top of the cliff to the boat is 15 degrees, how tall is the cliff?
Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
A boat is 1000 meters from a cliff. If the angle of depression from the top of the cliff to the boat
is 15 degrees, how tall is the cliff?
~~~~~~~~~~~~~~~~~~~~~~~~~~~

        The solution in the post by @mananth is fatally wrong.
        I came to bring a correct solution.

Angle of depression = 15 degrees
The sides containing the right angle
One side is the distance and the other side is the height
tan(15°) = height / distance from cliff
tan(15°) = height / 1000
height = 1000 * tan(15°)
= 1000 * 0.26794919243
267.95 meters is the height (rounded).         ANSWER

Solved correctly.

Question 271643: Find the cost of painting the outer surface of 24 oil containers at $2.50 per square meter, the dimensions of each container being 60 centimeters by 40 centimeters by 56 centimeters.
Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
Find the cost of painting the outer surface of 24 oil containers at $2.50 per square meter,
the dimensions of each container being 60 centimeters by 40 centimeters by 56 centimeters.
~~~~~~~~~~~~~~~~~~~~~~~~~

        As I read the solution in the post by @mananth, I was shocked.
        He writes "There are 8 faces in a cuboid 4 faces will have same area. The other four faces will have same arae."

        In opposite, a cuboid has 6 faces, and opposite faces have equal areas.

        So, the @mananth solution is a fatal nonsense.

        I came to bring a correct solution.

If a rectangular box has dimensions L (the length), W (the width) and H (the height), then its surface area is S = 2*(LW + LH + HL). In our case, L = 60 cm = 0.6m; W = 40 cm = 0.4 m; H = 56 cm = 0.56 m, and the surface area is 2*(0.6*0.4 + 0.6*0.56 + 0.40*0.56) = 2*0.8 = 1.6 m^2. So, the cost of the painting is $2.50 times 1.6, or 4 dollars. ANSWER

Solved correctly.

Question 63293: Rashad wants to wallpaper the four walls of his bedroom. the room is rectangular and measures 11 feet by 13 feet. the cieling is 10 feet high. a roll of wallpaper at a store is 2.5 feet wide and 50 feet long. how may rolls should he buy?
Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
Rashad wants to wallpaper the four walls of his bedroom. the room is rectangular and measures 11 feet by 13 feet.
the cieling is 10 feet high. a roll of wallpaper at a store is 2.5 feet wide and 50 feet long. how may rolls should
he buy?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        The solution in the post by other person is incorrect.
        See my correct solution below.

The surface area to cover by wall paper is the area of four walls

        11*10 + 13*10 + 11*10 + 13*10 = 480 ft^2.

The area of one roll is 2.5*50 = 125 ft^2.

Find the ratio = 3.84.

We should round this decimal, 3.84, to the closest GREATER integer number, which is 4.

ANSWER.   Rashad should buy 4 roll of wallpaper.

Solved correctly.

Question 262612: Clint is constructing two adjacent rectangular dog pens. Each pen will be three times as long as it is wide, and the pens will share a common long side. If clint has 65 ft of fencing, what are the dimensions of each pen?
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13367)   (Show Source):
You can put this solution on YOUR website!

For tutor @ikleyn....

You have a typo in your response that might confuse students. There are 4 "sides" of length x, not 3.

Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
Clint is constructing two adjacent rectangular dog pens. Each pen will be three times as long as it is wide,
and the pens will share a common long side. If clint has 65 ft of fencing, what are the dimensions of each pen?
~~~~~~~~~~~~~~~~~~~~~~~~~

        The solution in the post by @mananth is incorrect conceptually: he incorrectly setup his governing equation.
        I came to bring a correct solution.

Let x be width of the pens, in feet. Then their length is 3x feet, according to the problem. We have 3 long sides of the length 3x ft each, and 4 short sides of the length x ft each. So, the total length of all dimensions is 3*(3x) + 4x. Therefore, the equation for the total fence length is 3*(3x) + 4x = 65 feet. Simplify and find x 13x = 65, x = 65/13 = 5 feet. ANSWER. The dimensions of each pen are 5 ft x 15 ft.

Solved correctly.

--------------------------

Thanks to tutor @greenestamps for noticing my typo.
I just fixed it.

Question 1027518: A trench 5m�16m and 0.75m deep is dug. The earth taken out is spread uniformly to form a layer 12.5cm deep and 4m wide. What is the length of the layer?
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39838)   (Show Source):
You can put this solution on YOUR website!
Two equal volumes, one unknown dimension in one of the expressions
, using all meter units

compute this.

Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
A trench 5m x 16m and 0.75m deep is dug. The earth taken out is spread uniformly to form a layer 12.5cm deep
and 4m wide. What is the length of the layer?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        Calculations in the post by @mananth are incorrect.
        I came to bring a correct solution.

A trench 5m x 16m and 0.75m deep is dug.
Volume of Trench = 5*16*0.75 = 60 m^3
The earth taken out is spread uniformly to form a layer 12.5cm
deep and 4m wide. (cuboid)
4*0.125*L = 60 m^3
L = 60/(4*0.125)
length = 120 m         <<<---===     ANSWER

Solved correctly.

Question 33038: Can you please solve this for me? I have to solve this using factoring:
The sum of the squares of two consecutive negative even integers is 340. Find the integers.

Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13367)   (Show Source):
You can put this solution on YOUR website!

(1) Following the directions to solve the problem using factoring....

Let the smaller integer be x; then the larger integer is x+2. The sum of the squares is 340, and the integers are negative:

or

The integers are negative, so the smaller integer is x = -14 and the large integer is x+2 = -12.

ANSWERS: -14 and -12

(2) Being smart about how you use algebra....

Use the powerful "trick" shown by tutor @ikleyn -- instead of using x and x+2 for the two integers, use x-1 and x+1. Then

or

The answers have to be negative, so x is -13 and the two integers are x-1 = -14 and x+1 = -12.

As you can see, using this trick leads to an equation that is easily solved and does not require the use of factoring. That's the reason for using the trick (in this and similar problems).

(3) Solving the problem as quickly as possible -- as if you are taking a timed competitive exam.

Solve informally using logical reasoning and mental arithmetic.

Half of 340 is 170.

What are the two squares of even integers that are closest to and on opposite sides of 170? They are 12^2 = 144 and 14^2 = 196.

The answers have to be negative, so they are -14 and -12.

Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
.
Can you please solve this for me? I have to solve this using factoring:
The sum of the squares of two consecutive negative even integers is 340. Find the integers.
~~~~~~~~~~~~~~~~~~~~~~~~~

        I will solve it in as simple way as I can.

We are looking for two consecutive even integer numbers n and (n+2). I will start from the central integer number 'm' between n and (n+2), so that n = m-1, n+2 = m+1. Then my equation is (m-1)^2 + (m+1)^2 = 340, (m^2 - 2m + 1) + (m^2 + 2m + 1) = 340, 2m^2 + 2 = 340, 2m^2 = 340 - 2 = 338, m^2 = 338/2 = 169, m = +/- = +/- 13. We are looking for two consecutive negative numbers, so these numbers are -14 and - 12. ANSWER CHECK. (-14)^2 + (-12)^2 = 196 + 144 = 340. ! Precisely correct !

Notice that the other tutor reduced the problem to solution of a quadratic equation, but left the solution to you.

I solved the problem completely in a simplest way, practically mentally
to the end, without solving a quadratic equation.

Question 1137073: Two cones are similar in shape. The ratio of the diameters of their bases is 2:7. The radius of the smaller cone is 4.5 inches. A) find the radius of the larger cone B) write the ratio of the height of the smaller cone to the height of the larger cone.
Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
Two cones are similar in shape. The ratio of the diameters of their bases is 2:7.
The radius of the smaller cone is 4.5 inches.
A) find the radius of the larger cone
B) write the ratio of the height of the smaller cone to the height of the larger cone.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        The solution in the post by @manant is incorrect.
        I came to bring a correct solution.

(a) The ratio of the cones radii is the same as the ratio of their diameters = . Hence, the radius of the larger cone is r2 = = = inches = 15.75 inches. ANSWER (b) Since the cones are similar, the ratio of their heights is the same as the ratio of their diameters = . ANSWER

Solved correctly.

-------------------------------

Remember that @mananth is not a human - it is a computer code, instead,
which generates his output automatically in correct mode or in wrong mode,
but it does not know in which mode does it currently work, and even does not ask
this question to himself, since it is not programmed to ask such questions
and is not programmed to think, to self-check or to self-control itself.

In other words, when you get answers from @mananth, no one human is personally responsible
if they are correct or wrong.

Question 1154763: Raul is 537 ft from the world's tallest totem pole in Albert Bay, Canada. He decides to place a mirror on the ground between himself and the totem pole to use the angle of reflection to estimate the pole's height indirectly. he places the mirror at a spot that is 519 ft from the pole and backs up to his original position. if Raul is 6 ft tall, how tall does he calculate the pole to be? (draw a picture to help you solve it.)
Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
Raul is 537 ft from the world's tallest totem pole in Albert Bay, Canada. He decides
to place a mirror on the ground between himself and the totem pole to use the angle of reflection
to estimate the pole's height indirectly. he places the mirror at a spot that is 519 ft from the pole
and backs up to his original position. if Raul is 6 ft tall, how tall does he calculate the pole to be?
(draw a picture to help you solve it.)
~~~~~~~~~~~~~~~~~~~~~

In the post by @mananth, the final unit should be ft (feet), not meters.

Be aware.

Question 1179513: The vertices of a triangle are P(-6,1), Q(-2,-5) and R(8,1).
Find the equation of the perpendicular bisector of the side QR.

Find the slope of the median of the triangle that passes through point R.

Find the slope of the altitude of the triangle that passes through point Q.
Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
The vertices of a triangle are P(-6,1), Q(-2,-5) and R(8,1).
Find the equation of the perpendicular bisector of the side QR.
Find the slope of the median of the triangle that passes through point R.
Find the slope of the altitude of the triangle that passes through point Q.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        The solution by @mananth in his post is incorrect.

        It is easy to see, if to substitute coordinates of point P(-6,1) into
        his final equation y = -5/3x - 11. You will get then y = -1 instead of y = 1.

        His error is in arithmetic error on the way.

        Below is my correct solution.

The vertices of a triangle are P(-6,1), Q(-2,-5) and R(8,1).
Find the equation of the perpendicular bisector of the side QR.
slope formula
(y2-y1)/(x2-x1) Q(-2,-5) and R(8,1).
slope of QR = (1-(-5))/(8-(-2)) = 3/5
T a perpendicular line will have a slope -5/3 (negative reciprocal)
slope = -5/3 and passing through P (-6,-1)
Plug value of the slope and point (-6 ,-1 ) in
Y = m*x + b
1 = 10 + b
b = 1 - 10
b = -9
So the equation is
Y = -5/3*x -9         ANSWER

-----------------------------------

Solved correctly and accurately.

Question 1179642: 1.the classroom is 20 feet long and 30 feet wide.the principal decided that the tiles would look attractive in that class.if each tile is 24 inches long and 36 inches wide,how many tiles are needed to fill the classroom?
2.a rectangle is 4 times as long as it is broad.if the lenth is increased by 4 inches and the width is decreased by 1 inch,the area would be 60 squares inches.what were the dimensions of the original rectangle?
Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
1.the classroom is 20 feet long and 30 feet wide. the principal decided that the tiles would look attractive
in that class. if each tile is 24 inches long and 36 inches wide, how many tiles are needed to fill the classroom?
~~~~~~~~~~~~~~~~~~~~~~~~~~

        In his post, @mananth solved this problem comparing the area of one single tile with the area of the classroom.

        Although in this case it leads to correct answer, nevertheless it is not a correct way to solve
        and to construct arguments. In other words, his solution is badly designed.

        Below I place my correct and accurate solution.

With the dimension of the room in one direction of 20 feet, 20/2 = 10 tiles can be placed in this direction. With the dimension of the room in the other direction of 30 feet, 30/2 = 15 tiles can be placed in this direction. Hence, 10 x 20 = 200 tiles are needed to cover the floor of the classroom. ANSWER

Now the problem is solved correctly and accurately.

Why it is necessary to look in each dimension separately ?

To guarantee that the integer number of tiles fits in each dimension.

Question 1204853: In the diagram below, circle with centre O has a radius of 5 cm. Segment AT is tangent to the circle. AO = 13 cm, and AX = XY (this length is labeled m). Find the length of m.
https://ibb.co/6HKJNjR
Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
In the diagram below, circle with center O has a radius of 5 cm. Segment AT is tangent to the circle.
AO = 13 cm, and AX = XY (this length is labeled m). Find the length of m.
https://ibb.co/6HKJNjR
~~~~~~~~~~~~~~~~~~~~~~~~~~

            This problem is not difficult.

First of all, from triangle AOT, AT = = = = 12 cm. Next, consider triangle XTO. Its leg XT has the length (12-m) cm. It is the tangent segment to the circle O. Continue XO further to intersection with the circle O. You will get the long secant of the length m+5+5 = m + 10 cm. The outer part of this secant has the length m. Using well known property of the tangent segment, secant and its outer part, you can write this equation = m*(m+10), which is = m*(m+10). Simplify and find "m" 144 - 24m + m^2 = m^2 + 10m 144 = 10m + 24m 144 = 34m m = = cm = 4 cm. ANSWER

Solved.

Question 1206179: You are the sailmaker creating a pattern for your customer. Your customer wants a triangular sail with two colors such as yellow and white. Your customer wants the yellow part to have the sides as 3 meters, 6 meters, and 21 meters. If the longest side of the sail were 84 meters, how many meters of white cloth would you cut to complete the sail? Given: Upper side - 6 meters. Provide solution and Illustration
Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
You are the sailmaker creating a pattern for your customer. Your customer wants a triangular sail with two colors
such as yellow and white. Your customer wants the yellow part to have the sides as 3 meters, 6 meters, and 21 meters.
If the longest side of the sail were 84 meters, how many meters of white cloth would you cut to complete the sail?
Given: Upper side - 6 meters. Provide solution and Illustration
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

For your info - a triangle with the side lengths 3 meters, 6 meters and 21 meters
DOES NOT EXIST and CAN NOT EXIST, according to triangle inequalities.

So you better throw this " problem " to the closest garbage bin,
since it is TOTALLY DEFECTIVE and is rotten to the core.

Question 449831: a circle has a diameter with endpoints (5, -2) and (-13 -6) what are the coordinates of the center of the circle
Found 2 solutions by mccravyedwin, ikleyn:
Answer by mccravyedwin(421)   (Show Source):
You can put this solution on YOUR website!
If the coordinates of A and B are ( x1, y1) and ( x2, y2) respectively, then the midpoint, M, of AB is given by the following formula M =
(5,-2)(-13,-6)
x=(5-13 )/2,y=(-2~~-13~~)/2
x= -4,y= ~~-7.5~~

Correction to mananth(16949)'s solution. She used -13, which was x2 where she should have used -6, which was y2. So her corrected solution is

If the coordinates of A and B are ( x1, y1) and ( x2, y2) respectively, then the midpoint, M, of AB is given by the following formula M =
(5,-2)(-13,-6)
x=(5-13 )/2,y=(-2-6)/2
x= -4,y= -4

Edwin

Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
A circle has a diameter with endpoints (5, -2) and (-13 -6) what are the coordinates of the center of the circle
~~~~~~~~~~~~~~~~~~~~~~~~~~~

        The solution by @mananth is INCVORRECT.
        I came to bring a correct solution.

If the coordinates of A and B are (x1,y1) and (x2,y2) respectively, then the midpoint, M, of AB is given by the following formula M = (, ) x = , y = . ANSWER. x = -4, y = -4.

Solved correctly.

Question 446465: The vertices of the triangle are A(6,1), B(6,7) and C(10,7) show that the triangle is right angled and find it's sides?
Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
The vertices of the triangle are A(6,1), B(6,7) and C(10,7) show that the triangle is right angled and find it's sides?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

            It is so simple . . . - much simpler than you think.

The side AB is vertical line x = 6, since points A and B have the same x-coordinate 6. The side BC is horizontal line y = 7, since points B and C have the same y-coordinate 7. Therefore, triangle ABC is a right-angled triangle. The length of side AB is the difference y-coordinates points A and B |7-1| = 6 units. The length of side BC is the difference x-coordinates points B and C |10-6| = 4 units. The hypotenuse AC has the length = = units.

At this point, the problem is solved completely: all questions are answered.

You do not need to make complicated reasoning or complicated calculations, as @mananth does.

This problem teaches you to retrieve out geometric information from coordinates
of given points in coordinate plane.

Question 437230: A kite flying 20 ft. above the ground is attached to a string 80 ft. long. The string is being held by a person on the ground.if the kite fell vertically to the ground, how far away from the stake would the kite land?
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13367)   (Show Source):
You can put this solution on YOUR website!

The use of trigonometry to solve the problem is a bit excessive -- unless a solution using trigonometry is required....

The problem simply involves a right triangle with hypotenuse 80 and one leg 20. The answer to the question is the length of the other leg, which can be found using the Pythagorean Theorem.

Use a calculator to find...

ANSWER: (to 2 decimal places) 77.46 feet

Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
A kite flying 20 ft. above the ground is attached to a string 80 ft. long.
The string is being held by a person on the ground. if the kite fell vertically to the ground,
how far away from the stake would the kite land?
~~~~~~~~~~~~~~~~~~~~

In his solution, @mananth uses this scheme

Find sin of the angle ---> find arcsin ---> find cos ---> find the horizontal distance.

In this scheme, one step is excessive.

The better scheme is

find sin(a) ---> find cos(a) = ---> find the horizontal distance.

I will use this improved scheme in my calculation

        (1)   Find sin(a) of angle of depression   sin(a) = = .

        (2)   Find cos(a) = = = 0.968245837.

        (3)   Horizontal distance = 80*cos(a) = 80*0.968245837 = 77.46 ft (approximately).         ANSWER

It is how this problem IS EXPECTED to be solved.

Question 418623: .
If the sum of the lengths of the edges of a cube is 48 inches, the volume of the cube is?

Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
If the sum of the lengths of the edges of a cube is 48 inches, the volume of the cube is?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        The solution and the answer in the post by @mananth both are incorrect.
        I came to bring a correct solution.

12 edges. 48/12 = 4. The cube's volume is 4*4*4 = = 64 cubic inches. ANSWER

Solved correctly.

Question 729493: A picec of wood is 1/2 metre long how many cm does another piece measure which is one and a half times as long?
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39838)   (Show Source):
You can put this solution on YOUR website!
Half meter is 50 centimeters. You know what to do to find and should not need to write steps on paper.

Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
A piece of wood is 1/2 metre long how many cm does another piece measure which is one and a half times as long?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Another piece is 75 centimeters long.

It is as clear as 2 x 2 = 4, and even more clear than that.

Question 1165192: Work out the next 5 terms in the number sequences below and explain your calculation using 'n' as your first term; e.g.n+1,n+2,n+3,etc:
a. 297;290;276;255;.....
b. 1;10;100;.....
c. 3;9;81;.....
d. 24;25;27;30;.....
Answer by CPhill(2264)   (Show Source):
You can put this solution on YOUR website!
Here are the next 5 terms for each number sequence, along with the explanation of the calculation using a term-to-term rule based on its position in the sequence, $n$.
---
## a. 297; 290; 276; 255; **227; 192; 150; 101; 45**
### Analysis:
This sequence is based on a decreasing difference that increases by a fixed amount each time (a quadratic sequence).
| Term ($T_k$) | Value | Difference ($\Delta_1$) | Second Difference ($\Delta_2$) |
| :---: | :---: | :---: | :---: |
| $T_1$ | 297 | - | - |
| $T_2$ | 290 | -7 | - |
| $T_3$ | 276 | -14 | **-7** |
| $T_4$ | 255 | -21 | **-7** |
| $T_5$ | **227** | -28 | **-7** |
| $T_6$ | **192** | -35 | **-7** |
| $T_7$ | **150** | -42 | **-7** |
| $T_8$ | **101** | -49 | **-7** |
| $T_9$ | **45** | -56 | **-7** |
### Next 5 Terms:
227, 192, 150, 101, 45
### Term-to-Term Rule:
The difference decreases by 7 each time, starting with 7.
* Next term ($n+1$): **$n - 28$**
* $n+2$: **$(n-28) - 35$**
* $n+3$: **$(n-28-35) - 42$**
* ...and so on.
---
## b. 1; 10; 100; **1,000; 10,000; 100,000; 1,000,000; 10,000,000**
### Analysis:
This is a geometric sequence where each term is 10 times the previous term.
### Next 5 Terms:
1,000, 10,000, 100,000, 1,000,000, 10,000,000
### Term-to-Term Rule:
The next term is the previous term multiplied by 10.
* Next term ($n+1$): **$n \times 10$**
* $n+2$: **$(n \times 10) \times 10$**
* $n+3$: **$(n \times 10 \times 10) \times 10$**
* ...and so on.
---
## c. 3; 9; 81; **6,561; 43,046,721; $1.85 \times 10^{15}$; $3.40 \times 10^{30}$; $1.15 \times 10^{61}$**
### Analysis:
This sequence is generated by squaring the previous term.
* $3^2 = 9$
* $9^2 = 81$
* $81^2 = 6,561$
### Next 5 Terms:
1. $81^2 = \mathbf{6,561}$
2. $6,561^2 = \mathbf{43,046,721}$
3. $43,046,721^2 = \mathbf{1,853,020,188,851,841} \quad (\approx 1.85 \times 10^{15})$
4. $(\approx 1.85 \times 10^{15})^2 \approx \mathbf{3.43 \times 10^{30}}$
5. $(\approx 3.43 \times 10^{30})^2 \approx \mathbf{1.17 \times 10^{61}}$
### Term-to-Term Rule:
The next term is the square of the previous term.
* Next term ($n+1$): **$n^2$**
* $n+2$: **$(n^2)^2 = n^4$**
* $n+3$: **$((n^2)^2)^2 = n^8$**
* ...and so on.
---
## d. 24; 25; 27; 30; **34; 39; 45; 52; 60**
### Analysis:
This sequence is based on an increasing difference (an arithmetic progression of differences).
| Term ($T_k$) | Value | Difference ($\Delta_1$) | Second Difference ($\Delta_2$) |
| :---: | :---: | :---: | :---: |
| $T_1$ | 24 | - | - |
| $T_2$ | 25 | +1 | - |
| $T_3$ | 27 | +2 | **+1** |
| $T_4$ | 30 | +3 | **+1** |
| $T_5$ | **34** | +4 | **+1** |
| $T_6$ | **39** | +5 | **+1** |
| $T_7$ | **45** | +6 | **+1** |
| $T_8$ | **52** | +7 | **+1** |
| $T_9$ | **60** | +8 | **+1** |
### Next 5 Terms:
34, 39, 45, 52, 60
### Term-to-Term Rule:
The difference increases by 1 each time, starting with 1.
* Next term ($n+1$): **$n + 4$**
* $n+2$: **$(n+4) + 5$**
* $n+3$: **$(n+4+5) + 6$**
* ...and so on.

Question 730259: A doorway is 77 inches tall and 36 inches wide. What is the length of the diagonal
from the top left corner to the bottom right corner of the doorway?

Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
A doorway is 77 inches tall and 36 inches wide. What is the length of the diagonal
from the top left corner to the bottom right corner of the doorway?
~~~~~~~~~~~~~~~~~~~~~~~

It is = 85 inches (precisely). ANSWER

Solved.

Question 732119: a street sign at the end of Dion's street is 8feet high. At a certain time, it casts a shadow of 6 feet long. Dions tied a rope to the top of the sign and fastened it in the ground at the end of the shadow, making a triangle with the sign, the rope, and the shadow. what is the length of the rope

Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
A street sign at the end of Dion's street is 8feet high. At a certain time, it casts a shadow of 6 feet long.
Dion tied a rope to the top of the sign and fastened it in the ground at the end of the shadow, making a triangle
with the sign, the rope, and the shadow. what is the length of the rope
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Apply the Pythagorean theorem and get the length of the rope = = = = 10 feet. ANSWER

Solved.

Question 732251: A ball is tossed upward with an initial velocity of 122 ft/s from a platform that is 700 ft above the surface of the earth. After t seconds, the height of the ball above the ground is given by the equation h = -16t^2 + 122t + 700. What is the maximum height of the ball? Round to the nearest tenth of a foot.
Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
A ball is tossed upward with an initial velocity of 122 ft/s from a platform that is 700 ft above
the surface of the earth. After t seconds, the height of the ball above the ground is given
by the equation h = -16t^2 + 122t + 700. What is the maximum height of the ball?
Round to the nearest tenth of a foot.
~~~~~~~~~~~~~~~~~~~~~~

This given equation h = -16t^2 + 122t + 700 has the leading coefficient negative, -16. So, it describes a parabola opened downward. Such a parabola has a maximum. According to the general theory, a parabola y = ax^2 + bx + c with a negative leading coefficient 'a' has a maximum at the point x = . In our case, the maximum is achieved at x = = = 3.8125. It means that the maximum height is achieved at 3.8125 second after tossing. To find the maximum height h, substitute x = 3.8125 into the formula and calculate h = -16*3.8125^2 + 122*3.8125 + 700 = 932.6 feet (rounded as requested). ANSWER. The maximum height of the ball is 932.6 feet.

Solved.

Question 732950: markita bought a roll of carpet that is 8.5 feet wide. The maximum area the carpet will cover is 136 square feet. What is the longest possible length of the carpet?
Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.

To find the length, divide the area by the width.

It is as clear as 2 x 2 = 4, and does not require to submit to the forum.

Question 733677: hi i need a line that is 125% long
Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.

The length of a line is not measured in percents.

The measure of the length is centimeter, meter, millimeter, kilometer,
inch, foot, yard, mile etc.

So, think carefully and formulate correctly what you really want to get.

Question 739999: in the diagram ABC and AED are straight lines
BE and CE are parallel.Angles BAE=32 and angle EDC=68
work out the value of p
pls help me GOD will help u

Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
in the diagram ABC and AED are straight lines
BE and CE are parallel. Angles BAE=32 and angle EDC=68
work out the value of p
pls help me GOD will help u
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

As the problem is given in your post, it is absurdist and describes a situation,
which never may happen.

Indeed, the lines BE and CE can not be parallel, since they contain a common point E.

Also, the meaning of 'p', which you ask "to work out", is not defined in the post.

This problem is as far from to be a true Math problem, as the heaven is far from earth.

Learn to write your problems correctly.

Question 740073: A rectangular box has a base area of 12 square centimeters. The height of the box is 3 2/3 centimeters. What is the volume of the box?

Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
A rectangular box has a base area of 12 square centimeters. The height of the box is 3 2/3 centimeters.
What is the volume of the box?
~~~~~~~~~~~~~~~~~~~~~~~

The volume of the box is the product of the base area by the height volume = 12 * 3 2/3 = 12 * = 4*11 = 44 cm^3. ANSWER

Solved.