Questions on Word Problems: Geometry answered by real tutors!

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The area of the original sheet of 10X10 sheet of paper is:


sq.in.
The remaining area after the removal of the rectangular piece is given as:

so then, the area of the removed rectangular piece is just 100 - 80 = 20 sq.in.
The width of the removed piece is 4 inches, so, for the removed piece...



inches.

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Let's bisect the problem carefully.
First illustration:
and are supplementary angles, so -----> eqn 1
It shows is a supplement of and a exist that the supplement which is , is 12 degrees greater than 3 times the complement.
2nd illustration:
We make again and complementary angles this time, so
----> eqn 2. we make a complement of . To show the condition on the 1st illustration: ----> 12 greater than 3 times the complement, being a complement of right?
Substitute the to eqn 1:




For in question via eqn 1:

To check, go back to the :




Thank you,
Jojo

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L=6W
H=L/3
LWH=6144
6W*W*L/3=6144
6W*W*6W/3=6144
36W^3/3=6144
12W^3=6144
W^3=6144/12
W^3=512
W=CUBERT512
W=8 FOR THE WIDTH
L=6*8=48 FOR THE LENGTH
H=48/3=16 FOR THE HEIGHT.
PROOF:
8*48*16=6144
6144=6144

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Correct; Yes.
Cheers,
Stan H.

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If a coin is flipped 4 times, find P (exactly 3 heads).
-------------
Write down ALL of the possible results for flipping a coin 4 times;
there are 16 patterns.
--------
hhhh
--------
hhht
hhth
hthh
thhh
--------
tthh
thth
thht
htht
hhtt
htth
----
httt
thtt
ttht
ttth
------
tttt
=========
Note: that listing is called the "sample space" for the experiment
of tossing 4 coins as it lists all the possible results.
=========
Now count the number of coutcomes that have "exactly 3 heads".
=========
The probability of exactly 3 heads is the number of outcomes that
have 3 heads divided by the total number of possible outcomes.
==============================
Cheers,
Stan H.
==============================

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By Ratio & Proportion:
Let = # of copies:

, cross multiply

-------->

Thank you,
Jojo

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Area of Rectangle=

Area of square block=
Therefore,
--->#of blocks to fit
Thank you,
Jojo

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The length of a rectangle is 5 less than thrice its width. Find the dimensions if the area is 112 squared cm.
---------------------------
L = 3W - 5 (The length of a rectangle is 5 less than thrice its width)
L*W = 112
Sub for L
(3W-5)*W = 112
3w^2 - 5W - 112 = 0

Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)

Quadratic equation (in our case ) has the following solutons: For these solutions to exist, the discriminant should not be a negative number. First, we need to compute the discriminant : . Discriminant d=1369 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: 7, -5.33333333333333. Here's your graph:

Ignore the negative answer, so W = 7 (the online solver always uses x)
W = 7
L = 3*7 - 5 = 16

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Remember:
-------------------------------------> working eqn
But , 5 less thrice the width
So substitute,
, distribute


By Quadratic: , , &




, It has 2 values:
--------------------> TO BE USED , WIDTH
-----------------> DON'T USED (-)
Therefore,


In doubt? Go back working eqn:



Thank you,
Jojo

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Let x be the width, then the length is .
The perimeter = 2*width + 2*length
=
As the perimeter is 60, we have:

Solving for x, we have




So
the width is 14 cm,
the length is .

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3X*X=75
3X^2=75
X^2=75/3
X^2=25
X=SQRT25
X=5 ANSWER FOR THE WIDTH.
3*5=15 ANSWER FOR THE LENGTH.
[Proof:]
15*5 = 75
What is your problem? You have a valid proof.

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let x be the measure of the first angle.
Then the second angle is x - 30, and the third angle is 2(x-30).
Their sum is x + (x - 30) + 2(x - 30)
As the three angles of any triangle add up to 180 degrees, we have:
x + (x - 30) + 2(x - 30)=180
Solving for x, we have:
x + x - 30 + 2x - 60 = 180
4x -90 = 180
4x = 270
x = 67.5
So
The first angle is 67.5 degrees
The second angle is x-30 = 67.5-30=37.5 degrees
The third is twice the second. so it is 2*37.5 = 75 degrees

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Let x be the length of the shortest side.
Then the lengths of the other two sides are x + 4 and 2x.
The perimeter of the triangle is x + (x + 4) + 2x.
As the perimeter is 40, we have
x + (x + 4) + 2x = 40
Solving for x, we have
x+ x + 4 + 2x = 40
4x + 4 = 40
4x = 36
x = 9
So the length of the shortest side is 9 cm.
The lengths of the other two sides are:
x + 4 = 9 + 4 = 13 cm
2x = 2*9 = 18 cm

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The lefthand splits the pole's length into two pieces. One piece is twice as long as the other.
The full length is 15.
2/3 is below the lefthand, 1/3 above it.

So there her left hand is 10 feet from the far end of the pole.
The right hand is 1.5 feet above the left. So the right hand is at
from the far end.

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Make a table of x,f(x) data points.
Then plot the points and connect the points.
-2 ,25
-1 ,5
0 ,1
1 ,0.2
2 ,0.04
3 ,0.008
As you can see from the points as x grow large negatively, f(x) increases quickly.
Also, as x grows large positively, f(x) quickly goes to zero.

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y=f(x)
f(x)=7x

First plug in a number for x. Maybe 1 first:
f(1)=7

So one point is (1,7).

Now plug in 2:
f(2)=14

So another point is (2,14).

Now plug in 3:
f(3)=21

So another point is (3,21).

Keeping plugging in numbers this way and plot them on a graph. Remember to plug in negative numbers too, such as:

f(-1)=-7
(-1,-7)

I hope this helps!

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f(x)=4x-3
(graph 300x200 pixels, x from -6 to 5, y from -10 to 10, 4x -3).

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The level of thorium in a sample decreases by a factor of one-half every 4.2 million years. A meteorite is discovered to have only 7.6% of its original thorium remaining. How old is the meteorite?
-------------------------
A(t) = Ao(1/2)^(t/4.2 mil)
This says "the amount at time t A(t) is the amount you start with (Ao)
multiplied by (1/2) (t/4.2 mil) times.
-------------
0.076Ao = Ao*(1/2)^(t/4.2 mil)
0.076 = (1/2)^(t/4.2 mil)
Take the log of both sides to get:
log 0.076 = (t/4.2 mil)*log(1/2)
t/4.2 mil = log0.076/log0.5
t = 4.2 mil * 3.717857...
t = 15.614998.. million years old.
================
Cheers,
Stan H.

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let x="width of path"

area of garden is 18*13 or 234 __ area of garden plus path is 234+516 or 750

(18+2x)(13+2x)=750 __ 234+62x+4x^2=750 __ dividing by 2 __ 2x^2+31x+117=375 __ subtracting 375 __ 2x^2+31x-258=0

factoring __ (2x+43)(x-6)=0

2x+43=0 __ x=-43/2 __ negative value not realistic

x-6=0 __ x=6

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end of 1st yr __ 10000(1+r)

end of 2nd yr __ [10000(1+r)+3500](1+r)=15569.75

10000+10000r+3500+10000r+10000r^2+3500r=15569.75 __ subtracting 15569.75 __ 10000r^2+23500r-2069.75=0

use quadratic formula to find r


r=8.5%

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If x=1 and x=-8 are zeros for a quadratic equation, then they satisfy,


The simplest quadratic equation would be the two left hand sides multiplied together.

Using the FOIL method we can expand the left hand side,

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An express and local train leave Gray’s Lake at 3 P.M. and head for Chicago 50 miles away. The express travels twice as fast as the local, and arrives 1 hour ahead of it. Find the speed of each train.
;
Let s = speed of the local train
then
2s = speed of the express
:
Write a time equation: Time =
:
Local time - express time = 1 hr
- = 1
Multiply equation by 2s to get rid of the denominators
2s* - 2s* = 2s(1)
Results
2(50) - 50 = 2s
:
100 - 50 = 2s
s =
s = 25 mph speed of the local, obviously, 50 mph is speed of express
:
:
Check solution in our time equation:
- = 1
2 - 1 = 1
:
:
Pretty easy, right?

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Rate times time equals distance.
The express train travels at a rate of and takes to travel the 50 miles.
1.
The local train travels at a rate of and takes to travel the 50 miles.
2.
You also know that the rate of the express train is twice the rate of the local train.

And finally you know that the local train arrived 1 hour after the express train.

From 2,
2.
Substitute for and
2.



From 1, you know that



The express train travels at 50 miles per hour.
From 1,



The express train takes 1 hour to reach Chicago.



The local train travels 25 miles per hour.



The local train takes 2 hours to reach Chicago.

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let d=dimes, n=nickels

"Joe has a collection of nickels and dimes that is worth $5.65" __ 10d+5n=565

"If the number of dimes were doubled and the number of nickels were increased by 8, the value of the coins would be $10.45"
__ 10(2d)+5(n+8)=1045 __ 20d+5n+40=1045 __ 20d+5n=1005

subtracting equations __ (20d+5n)-(10d+5n)=1005-565 __ 10d=440 __ d=44

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--------------------> eqn 1
But, , so

, adding & cross multiply

--------->

And,
In doubt? Go back eqn 1:



Thank you,
Jojo

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G'day Sally;
Let x = the amount invested at 8% and the remainder ($9000-x) is invested at 9%
The amount of interest earned on these amounts can be expressed as:
x(0.08) This is amount earned at 8%.
($9000-x)(0.09) This is the amount earned at 9%
The sum (+) of these two amounts is given as $770.00, so you can write the equation to solve for x, the amount invested at 8%
(0.08)x + (0.09)($9000-x) = $770 Simplify and solve for x.
0.08x + $810 - 0.09x = $770 Subtract $810 from both sides.
0.08x-0.09x = -$40 Combine the x's on the left side.
-0.01x = -$40 Finally, divide both sides by -0.01 to get x by itself.
x = $4000
So $4000 was invested at 8% and $9000-$4000 = $5000 was invested at 9%
Let's check the solution:
0.08($4000) + 0.09($5000) = $320 + $450 = $770 The total interest earned.

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Use the fact that the sum of the interior angles of any triangle is equal to 180 degrees to set up an equation
Here, < A=90, < B=2x and < C=3x,their sum is 90+2x+3x.
Setting it equal to 180,we have
90+2x+3x=180
Solving for x,we obtain:
90 + 5x = 180
5x = 90
x = 18
therefore
< B= 2x = 2*18 = 36 degrees
< C = 3x = 3*18 = 54 degrees.

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Think about this!
Through any three non-collinear points, there is exactly one plane!
The planes that you identify here must include the point E, so you can list all of the possible combinations of the points, A, B, C, D, with the point E.
E,A,B
E,A,C
E,A,D
E,B,C
E,B,D
E,C,D
So it seems that there are 6 distinct planes determined by the five points if point E is to be included in each one.

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All of your answers look good except for the first problem, part e.
Given: ...You said "This is not a polynomial" Why is it not?
It certainly qualifies as a polynomial (a trinomial in this case) because there are no terms with variables in the denominator and no terms with variables under a radical sign.
This polynomial is of degree 4 because of the

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Same as Problem#151709
See ANSWER# 111525
Thank you,
Jojo

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A function gives the value of C as 2 × (22/7) × r. Find C when r = 21 cm and r = 84 cm.

C has 2 values:
1st,


2nd,


Thank you,
Jojo

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A rectangular parking lot is 100 ft longer than it is wide. Determine the dimensions of the parking lot if it measures 500 ft diagonally.
.
Let x = width
x+100 = length
.
Applying Pythagorean theorem:
x^2 + (x+100)^2 = 500^2
x^2 + x^2 + 200x + 10000 = 250000
2x^2 + 200x - 240000 = 0
x^2 + 100x - 120000 = 0
.
Factoring we get:
(x-300)(x+400) = 0
x = {-400, 300}
Only the positive answer makes sense so
x = 300 feet (width)
.
x+100 = 400 feet (length)
.
Conclusion:
parking lot is 300 feet by 400 feet

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A lawyer bills her clients $200 per hour of service. If a client's case requires 39 hours to complete, use proportion to calculate how much the client will owe the lawyer (excluding tax).
.
From: "A lawyer bills her clients $200 per hour of service." we get the ratio:
$200:1 hour
.
From: "a client's case requires 39 hours to complete" we get the ratio:
Let x = how much the client owes
x: 39 hours
.
Therefore we have:
200/1 = x/39
Cross multiplying we get:
200 * 39 = x
$7800 = x

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1st degree because y is to the power of one.

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right

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b. (11y2)1/2 + 14 (degree 2)
c. 10 + (19)1/2x (is not a polynomial)
d. 2 + 15x (degree 1)
e. 52y4 + 7x + 2 (is not a polynomial)
f. (68)1y1
g. x3 + 3x – 9 (degree 3)
h. (2x)1/2 + 4x - 8 (degree 1)

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The problem gives you a "function of x":
f(x) = 4x3 - 3x2 - x + 2
.
Now, to find f(1), it simply asks you to find the value of f(x) when x=1
.
So, simply substitute in 1 wherever you see the x and solve:
f(x) = 4x^3 - 3x^2 - x + 2
Substituting we get:
f(1) = 4(1)^3 - 3(1)^2 - 1 + 2
f(1) = 4(1) - 3(1) - 1 + 2
f(1) = 4 - 3 - 1 + 2
f(1) = 6 - 4
f(1) = 2 (this is what they want)

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Answer: 20% of surface area of each is not painted
Proof:
Before it was cut into half, all the sides are equal right? (a cube is a square with 3-D body). Then it was cut to half and became a rectangular solid. One side of which is not painted.
We have to remember of a rectangular solid:
---------------------> EQN 1
Important: before it has ALL SIDES EQUAL, and when cut into half there appears now the and the ----> the dimensions of the now will be half of the : , Also the remains the same as the . Only the changed when cut into half.
For the of the new rectangular solid, since one side is NOT painted, -------------------> EQN 2
NOW, IF YOU PLUG IN VALUES IN EQN 1 AND EQN 2, REMEMBER , and make . Get ABSOLUTE VALUE (no negative)*100% ----------> PERCENTAGE of Surface Area NOT painted.
Hope this helps.
Thank you,
Jojo

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We take the 1st condition: a. The measure of angle CBD is 10 less than twice the measure of angle ABD.
-----------------------> eqn 1
Since is in the interior of angle , there forms an angle right? Therefore, angle , see the correlation there?
It's better you draw it to follow the interpretation better.
Since , then ---------------------> eqn 2
.
Equating the 2 eqns:




Go back eqn 2:

-----------------> FINAL ANSWER
IN doubt? go back eqn 1:


cool,
Thank you,
Jojo

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You have the correct answers.
You can easily check you answers by just plugging the value for x and y into the original two equations and verifying the result is "true" (that is 12=12 and 3=3)

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degree is based on the power of the largest variable (or is degree 0 if everything is constants).
You have an expression that contains y to a power of 1. So the degree is 1

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Let's start with the formulas for the area and the perimeter of a rectangle.


The perimeter is given as 250 feet, so we can write:
Dividing both sides by 2, we get:
Rewrite this as:
and substitute it into the formula for the area (A = L*W),
Simplifying this, we get:
Now this is a quadratic equation and if you were to graph this, you see a parabola that opens downward, thus there would be a maximum value of A (also known as the vertex).
You can find the value of W at this vertex by:
This comes from the standard form for a quadratic equation:
In this problem, a = -1 and b = 125, so, making the appropriate substitutions, we get:

Now this is the value of W (the width) that would make the area (A) a maximum.
The length (L) is:



So the length and the width of the rectangle would be L = 62.5 feet and W = 62.5 feet. In other words, the rectangle is a square.

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Let us see the situations:
1st customer:
5MP+8HP=$840 ----------------> eqn 1
2nd customer:
3MP+4HP=$480 ----------------> eqn 2
In eqn 2 we get, and substitute in eqn 1:



----------------------> cost of an MP3 player
For Head phones set, go back eqn 2:



In doubt? Go back eqn 1:



Thank you,
Jojo