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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(10806)   (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(53748)   (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(53748)   (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(53748)   (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(53748)   (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(13327)   (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(53748)   (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(39792)   (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(53748)   (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(13327)   (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(53748)   (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(53748)   (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(53748)   (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(53748)   (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(53748)   (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(53748)   (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(53748)   (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(53748)   (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(53748)   (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(13327)   (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(53748)   (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(53748)   (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(39792)   (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(53748)   (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(2189)   (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(53748)   (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(53748)   (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(53748)   (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(53748)   (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(53748)   (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(53748)   (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(53748)   (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.

Question 744559: A square region with a side length of 2 inches lies within another, larger square region as shown above. If the larger square region has an area twice that of the smaller square region, then how much larger is the diagonal of the large square region than the diagonal of the small square region?
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
A square region with a side length of 2 inches lies within another, larger square region as shown above.
If the larger square region has an area twice that of the smaller square region, then how much larger
is the diagonal of the large square region than the diagonal of the small square region?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Both squares are similar figures geometrically (as any two squares). We know that the area ratio is 2 (the greater to the smaller). Hence, the similarity coefficient (linear) is , the greater to the smaller. It means that the diagonal of the greater square is times as long as that of the smaller square. The diagonal of the smaller square is = = cm. Hence, the diagonal of the greater square is = 2*2 = 4 cm. To determine, how much it is longer, subtract the two numbers = 1.1716 cm (rounded). ANSWER

Solved.

Question 574963: a pie-shaped (triangular) lake-front lot has a perimeter of 1200 ft. One side is 200 ft longer than the shortest side, while the third side is 400 feet longer than the shortest side. What are the lengths of all three sides?

Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.

In his post, tutor @Theo obtained the solution for the side lengths of the "triangle":
200 ft, 400 ft and 600 ft.

His solution is correct, but he didn't noticed that this triangle is degenerated and, actually,
is not a triangle, since the triangle inequalities are not held.

So, I am warning a reader, that this problem is defective, since it produces
a degenerated triangle, which is not a triangle, at all.

Question 477556: A square has an area of x^2 + 6x + 9 find the length of a side. Make a sketch of the square Please help
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
A square has an area of x^2 + 6x + 9. Find the length of a side. Make a sketch of the square Please help
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        Tutor @Theo' solution is many words.

        I don't like toooo wordy solutions: I think that too wordy explanations are bad style to teach Math.
        So, I will write another solution, much shorter and, I believe, more educative.

The expression x^2 + 6x + 9 represents an algebraic perfect square . So, the area of the square is square units. Hence, the side length of this square is square root of this perfect square . So, our first desire is to declare that the side length is (x+3) units. But it would be too hastily and not always correct. The reason is that this expression (x+3) can be negative, since we don't know what 'x' is. Meanwhile, the length of the square must be positive due to the meaning of this notion/conception. Therefore, more accurate is to write = |x+3| , using the absolute value. This form is universally correct for all possible values of 'x'. ANSWER. If the area of a square is x^2 + 6x + 9 square units, then the side of the square is |x+3| units.

Solved, with complete explanations in compact form.

Question 488212: The perimeter of this block of land is 72 m. Find:
a the length of the block
b the area of the block.
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
The perimeter of this block of land is 72 m. Find:
(a) the length of the block
(b) the area of the block.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

This problem is incomplete and can not be solved/answered.

Before submitting to this forum, read your posts attentively
and prepare them as accurately as it should be done.

Question 495218: Please help me solve the word problem using
The Rational Zero Theorem
An open metal tank is to be made from a rectangular piece of stainless steel that measures 12 by 6 feet, by cutting out squares of the same size from each corner and bending up the sides. If the volume of the tank is to be 50 ft^3 , how large the square should be cut from each corner?
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
Please help me solve the word problem using
The Rational Zero Theorem
An open metal tank is to be made from a rectangular piece of stainless steel that measures 12 by 6 feet,
by cutting out squares of the same size from each corner and bending up the sides.
If the volume of the tank is to be 50 ft^3 , how large the square should be cut from each corner?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

I plotted the graph   V(x) = (12-2x)*(6-2x)*x   for the volume as a function of the size x
of the cut squares at the corners.

I used free of charge online plotting tool www.desmos.com/calculator

You can see the plot under this link
https://www.desmos.com/calculator/9ie79ro4n0
https://www.desmos.com/calculator/9ie79ro4n0

The plot shows that the maximum possible volume is about 41.57 cubic feet
and can not be more than that.

So, your problem is posed INCORRECTLY: the volume of 50 ft^3 CAN NOT be reached.

As presented in the post, this problem is TOTALLY FALSE and FATALLY DEFECTIVE.

            A reprimand to the creator of this task.

Question 556812: ANGLE ABE AND ANGLE CBD ARE VERTICAL ANGLES AND BOTH ARE COMPLEMENTARY WITH ANGLE FGH, IF m ANGLE ABE = (3x-1) AND m ANGLE FGH = (4x), what is m ANGLE CBD?
THIS IS THE CORRECT QUESTION..MESSED UP THE ONE BEFORE THIS
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
ANGLE ABE AND ANGLE CBD ARE VERTICAL ANGLES AND BOTH ARE COMPLEMENTARY WITH ANGLE FGH,
IF m ANGLE ABE = (3x-1) AND m ANGLE FGH = (4x), what is m ANGLE CBD?
~~~~~~~~~~~~~~~~~~~~~~~~

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

if angle ABE and CBD are vertical angles, then their measure are equal.
if angle ABE is equal to (3x-1) and angle FGH = (4x), then angle CBD must also be equal to (3x-1).
to find the value of x, we use the equation of angle ABE + angle FGH = 90 degrees because they are complementary.
since ABE = 3x-1 and FGH = 4x, this means that:

3x-1 + 4x = 90 degrees

combine like terms to get:
7x-1 = 90

add 1 to both sides of the equation to get:
7x = 91

divide both sides of the equation by 7 to get:
x = 91/7 = 13.

angle FGH is equal to 4x which makes it equal to 4*13 = 52 degrees.
angle ABE is equal to 3x-1 which makes it equal to 3*13-1 = 38 degrees.
add them together to get 90 degrees

the numbers check out.

angle ABE is equal to 38 degrees
angle CBD is equal to angle ABE and so is equal to the same.

ANSWER. Angle CBD is 38 degrees..

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

Solved correctly.

Question 1165224: Miles takes a 350-mile round-trip flight to visit his parents. To qualify for Gold status at Awesome Airlines, one must fly at least 4200 and less than 15400 miles each year. How many times would Miles need to visit his parents each year to attain Gold status? Express your answer in interval notation.
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.

Calculate two fractions, and .

After completing these calculations, turn on your mind.

Question 1168317: The length of the base of the triangular sheet of canvass above the door of the tent is 3 ft. more than its height (altitude). The area is 5 ft2. Find the height and the length of the base of the triangle.
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
The length of the base of the triangular sheet of canvass above the door of the tent is 3 ft. more
than its height (altitude). The area is 5 ft2. Find the height and the length of the base of the triangle.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

In this problem, you may forget about the tent, about the canvass and about the door, since they are not relevant to the problem. What is relevant, is the shape of the figure, which is a triangle. Let x be the altitude of the triangle (in feet); then the base of this triangle is (x+3) feet. Write the area equation for the triangle = 5 square feet, or x*(x+3) = 5*2 = 10 square feet. From this equation, you can guess the solution mentally: it is x = 2 feet. ANSWER. The altitude of the triangle is 2 ft; the base of the triangle is 5 ft.

Solved.

Question 1170888: The triangles ABC and XYZ are similar. The side AB is 5 cm long. The sides BC and
YZ are 7 cm and 3 cm longer than the side XY, respectively. Find the length of XY.
Found 2 solutions by MathTherapy, ikleyn:
Answer by MathTherapy(10806)   (Show Source):
You can put this solution on YOUR website!

The triangles ABC and XYZ are similar. The side AB is 5 cm long. The sides BC and YZ are 7 cm and 3 cm longer than the side XY, respectively. Find the length of XY. . Let XY = z With the 2 Δs being SIMILAR, AB and BC correspond to XY and YZ, respectively Since BC is 7 cm longer than XY, then BC = XY + 7, or z + 7 Also, since YZ is 3 cm longer than XY, then YZ = XY + 3, or z + 3 With BC being LONGER (XY + 7, or z + 7) than YZ (XY + 3, or z + 3), obviously ΔABC is the LARGER of the 2 <==== Using the former SIMILARITY-PROPORTION ---- Cross-multiplying (z - 3)(z + 5) = 0 z(XY) = 3 cm OR z = - 5 (ignore)

Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!

Question 1209920: It has been done before on this website but the question was wrong:
Two identical squares with side of (1+sqrt(2))m overlap to form a regular octagon. What is the area of the octagon?
Answer by greenestamps(13327)   (Show Source):
You can put this solution on YOUR website!

When the two squares overlap to form a regular octagon, the regions inside the squares and outside the octagon are eight isosceles right triangles.

Let x be the side length of each of those triangles; then x*sqrt(2) is the length of the hypotenuse, which is the side length of the octagon.

The side length of each square is then 2x+x*sqrt(2). Since the side length of the square is 1+sqrt(2),

The side length of the octagon, x*sqrt(2), is then

The area of a regular octagon with side length s is

The side length of our octagon is 1, so the area is

ANSWER:

Question 1173661: Triangle A, B and C are three points on a horizontal field. A is due west of B, the bearing of B from C is 125 degrees, AB = 430m and BC = 460m.
At a certain instant, a hot air balloon is at a point which is directly above C.Given that the angle of elevation of the hot air balloon from B is 5.2 degrees, find the angle of elevation of the hot air balloon from A.
Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website!
Let's break this problem down step-by-step.
**1. Draw a Diagram**
* Draw a horizontal line segment AB, with A to the left of B.
* Draw a line segment BC, where the bearing of B from C is 125 degrees. This means the angle between the north line from C and the line CB is 125 degrees. Since A is west of B, the angle ACB is 180 - 125 - 90 = 35 degrees.
* Draw a vertical line segment from C upwards to represent the hot air balloon's position. Let's call this point D.
* Connect BD and AD.
**2. Use the Law of Cosines to Find AC**
* We know AB = 430m, BC = 460m, and angle ACB = 35 degrees.
* Use the law of cosines: AC² = AB² + BC² - 2(AB)(BC)cos(ACB)
* AC² = 430² + 460² - 2(430)(460)cos(35°)
* AC² = 184900 + 211600 - 395600cos(35°)
* AC² = 396500 - 324484.7
* AC² = 72015.3
* AC = √72015.3 ≈ 268.36 m
**3. Find the Height of the Hot Air Balloon (CD)**
* The angle of elevation of the hot air balloon from B is 5.2 degrees.
* We have a right triangle BCD.
* tan(5.2°) = CD / BC
* CD = BC * tan(5.2°)
* CD = 460 * tan(5.2°)
* CD ≈ 460 * 0.0909
* CD ≈ 41.81 m
**4. Find the Angle of Elevation from A**
* We have a right triangle ACD.
* We know AC ≈ 268.36 m and CD ≈ 41.81 m.
* tan(angle CAD) = CD / AC
* tan(angle CAD) = 41.81 / 268.36
* tan(angle CAD) ≈ 0.1558
* angle CAD = arctan(0.1558) ≈ 8.85 degrees
**Therefore, the angle of elevation of the hot air balloon from A is approximately 8.85 degrees.**