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ACT Math : Solid Geometry

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Example Questions

Example Question #972 : Act Math

Sturgis is in charge of designing a new exhibit in the shape of a rectangular prism for a local aquarium. The exhibit will hold alligator snapping turtles and needs to have a volume of $150\:m^3$ . Sturgis knows that the exhibit will be $15\:m$ long and go $5\:m$ back into the wall.

What will the height of the new exhibit be?

Possible Answers:

$1.5\:m$

$2.2\:m$

$1\:m$

$4\:m$

$2\:m$

Correct answer:

$2\:m$

Explanation:

This sounds like a geometry problem, so start by drawing a picture so that you know exactly what you are dealing with.

Because we are dealing with rectangular prisms and volume, we will need the following formula:

Volume=length*width*height

V=l*w*h

We are solving for height, so you can begin by rearranging the equation to get by itself:

$\frac{V}{l*w}=h$

Then, plug in our knowns (, and )

$h=\frac{150m^3}{15m*5m}=\frac{150m}{75}=2\:m$

Here is the problem worked out with a corresponding picture:

Prism

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Example Question #1 : How To Find The Length Of An Edge Of A Prism

If three-quarters of the exhibit's volume will be water, how high up the wall will the water come?

Possible Answers:

$112.5\:m$

$1.0\:m$

$1.25\:m$

$1.5\:m$

Cannot be determined with the information provided

Correct answer:

$1.5\:m$

Explanation:

The trickiest part of this question is the wording. This problem is asking for the height of the water in the exhibit if the exhibit is three-quarters full. We can find this at least two different ways.

1) The longer way requires that we begin by finding three quarters of the total volume:

$150\:m^3*\frac{3}{4}=112.5\:m^3$

Now we go back to our volume equation, and since we are again looking for height, we want it solved for :

V=l*w*h

Becomes

$\frac{V}{l*w}=h$

$h=\frac{112.5\:m^3}{15\:m*5\:m}=1.5\:m$

2) The easier way requires that we recognize a key detail. If we take three-quarters of the volume without changing our length or width, our new height will just be three-quarters of the total height. We can solve for the total height of the exhibit by using the volume equation and rearranging it to solve for :

$\frac{V}{l*w}=h$

At this point, we can substitute in our given values and solve for :

$\frac{150\:m^3}{15\:m*5\:m}=2\:m$

So, the total height of the exhibit is $2\:m$ . We can now easily solve for three-quarters of the total height:

$h{}'=h*\frac{3}{4}$

$h{}'=2\:m*\frac{3}{4}=1.5\:m$

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Example Question #2 : Prisms

A right, rectangular prism has has a length of 6cm , a width of 9cm , and a height of 12cm . What is the length of the diagonal of the prism?

Possible Answers:

$\small 3\sqrt{13}$

$\small 18\sqrt{2}$

$\small 13\sqrt{3}$

$\small 2\sqrt{61}$

$\small 3\sqrt{29}$

Correct answer:

$\small 3\sqrt{29}$

Explanation:

First we must find the diagonal of the prism's base ( $\small d$ ). This can be done by using the Pythagorean Theorem with the length ( $\small l$ ) and width ( $\small w$ ):

$\small l^2+w^2=d^2$

$\small 6^2+9^2=d^2$

$\small 36+81=d^2$

$\small 117=d^2$

$\small d=\sqrt{117}$

Therefore, the diagonal of the prism's base is $\small \sqrt{117}cm$ . We can then use this again in the Pythagorean Theorem, along with the height of the prism ( $\small h$ ), to find the diagonal of the prism ( $\small D$ ):

$\small d^2+h^2=D^2$

$\small (\sqrt{117})^2+12^2=D^2$

$\small 117+144=D^2$

$\small D^2=261$

$\small D=\sqrt{261}=\sqrt{9}\sqrt{29}=3\sqrt{29}$

Therefore, the length of the prism's diagonal is $\small 3\sqrt{29}cm$ .

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Example Question #2 : How To Find The Diagonal Of A Prism

What is the diagonal of a rectangular prism with a height of 4, width of 4 and height of 6?

Possible Answers:

5.7

8.2

Cannot be determined

7.2

Correct answer:

8.2

Explanation:

In order to solve this problem, it's helpful to visualize where the diagonal is within the prism.

Diagonal

In this image, the diagonal is the pink line. By noting how it relates to the blue and green lines, we can observe how the pink line is connected and creates a right triangle. This very quickly becomes a problem that employs the Pythagorean theorem.

The goal is essentially to find the hypotenuse of this sketched-in right triangle; however, only one of the legs is given: the green line, the height of the prism. The blue line can be solved for by understanding that it is the measurement of the diagonal of a 4x4 square.

Either using trig functions or the rules for a special 45/45/90 triangle, the blue line measures out to be $4\sqrt2$ .

The rules for a 45/45/90 triangle: both legs are "" and the hypotenuse is " $s\sqrt2$ ". Keep in mind, this is is only for isosceles right triangles.

Now that both legs are known, we can solve for the hypotenuse (diagonal).

a^2+b^2=c^2

$(4\sqrt{2})^2+(6)^2=c^2$

32+36=c^2

68=c^2

$\sqrt{68}=c$

${\color{Blue} 8.2}=c$

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Example Question #2 : Prisms

Find the diagonal of a right rectangular prism if the length, width, and height are 3,4, and 5, respectively.

Possible Answers:

$3\sqrt{2}$

$\sqrt{23}$

$4\sqrt2$

$5\sqrt2$

Correct answer:

$5\sqrt2$

Explanation:

Write the diagonal formula for a rectangular prism.

$d=\sqrt{L^2+W^2+H^2}$

Substitute and solve for the diagonal.

$d=\sqrt{3^2+4^2+5^2}= \sqrt{9+16+25}= \sqrt{50}= 5\sqrt2$

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Example Question #977 : Act Math

If the dimensions of a right rectangular prism are 1 yard by 1 foot by 1 inch, what is the diagonal in feet?

Possible Answers:

$\sqrt3$

$\frac{\sqrt{1306}}{3}$

$\frac{1}{9}$

$\frac{\sqrt{1441}}{12}$

$\frac{\sqrt3}{3}$

Correct answer:

$\frac{\sqrt{1441}}{12}$

Explanation:

Convert the dimensions into feet.

$1 \textup{ yard}= 3\textup{ feet}$

$\textup{1 inch }= \frac{1}{12}\textup{ foot}$

The new dimensions of rectangular prism in feet are: $3\times1\times \frac{1}{12}$

Write the formula for the diagonal of a right rectangular prism and substitute.

$d=\sqrt{L^2+W^2+H^2}$

$d=\sqrt{3^2+1^2+\frac{1}{12}^2}=\sqrt{9+1+\frac{1}{144}}=\sqrt{\frac{1440}{144}+\frac{1}{144}}$

$d=\sqrt{\frac{1441}{144}} =\frac{\sqrt{1441}}{12}\textup{ ft}$

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Example Question #1 : How To Find The Surface Area Of A Prism

David wants to paint the walls in his bedroom. The floor is covered by a $10\ ft \times 16\ ft$ carpet. The ceiling is $8\ ft$ tall. He selects a paint that will cover $75\ ft^2$ per quart and $300\ ft^2$ per gallon. How much paint should he buy?

Possible Answers:

2 gallons and 1 quart

1 gallon

3 quarts

1 gallon and 2 quarts

1 gallon and 1 quart

Correct answer:

1 gallon and 2 quarts

Explanation:

Find the surface area of the walls: SA_walls = 2lh + 2wh, where the height is 8 ft, the width is 10 ft, and the length is 16 ft.

This gives a total surface area of 416 ft². One gallon covers 300 ft², and each quart covers 75 ft², so we need 1 gallon and 2 quarts of paint to cover the walls.

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Example Question #1 : Non Cubic Prisms

A box is 5 inches long, 5 inches wide, and 4 inches tall. What is the surface area of the box?

Possible Answers:

$130\ in^2$

$120\ in^2$

$140\ in^2$

$100\ in^2$

$25\ in^2$

Correct answer:

$130\ in^2$

Explanation:

The box will have six total faces: an identical "top and bottom," and identical "left and right," and an identical "front and back." The total surface area will be the sum of these faces.

Since the six faces consider of three sets of pairs, we can set up the equation as:

$SA=2(\text{top})+2(\text{left})+2(\text{front})$

Each of these faces will correspond to one pair of dimensions. Multiply the pair to get the area of the face.

SA=2(lw)+2(wh)+2(lh)

Substitute the values from the question to solve.

SA=2(5*5)+2(5*4)+2(5*4)

SA=50+40+40

$SA=130\ in^2$

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Example Question #1 : How To Find The Surface Area Of A Prism

What is the surface area of a rectangular brick with a length of 12 in, a width of 8 in, and a height of 6 in?

Possible Answers:

None of the answers are correct

$576\ in^2$

$382\ in^2$

$432\ in^2$

$216\ in^2$

Correct answer:

$432\ in^2$

Explanation:

The formula for the surface area of a rectangular prism is given by:

SA = 2LW + 2WH + 2HL

SA = 2(12 * 8) + 2(8 * 6) + 2(6 * 12)

SA = 2(96) + 2(48) + 2(72)

SA = 192 + 96 + 144

SA = 432 in²

216 in² is the wrong answer because it is off by a factor of 2

576 in³ is actually the volume, V = L * W * H

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Example Question #1 : How To Find The Volume Of A Prism

A box's length is twice as long as its width. Its height is the sum of its length and its width. What is the volume of this box if its length is 10 units?

Possible Answers:

750 units cubed

500 units cubed

150 units cubed

250 units cubed

300 units cubed

Correct answer:

750 units cubed

Explanation:

The formula for the volume of a rectangular prism is V = L*W*H , where "" is volume, "" is length, "" is width and "" is height.

We know that L = 2W and H = L+W . By rearranging L=2W , we get $W=\frac{L}{2}$ . Substituting $\frac{L}{2}$ into the volume equation for and L+W into the same equation for , we get the following:

$V = L \cdot \left(\frac{L}{2}\right)\cdot (L+\left(\frac{L}{2}\right))$

$V = L\cdot \left(\frac{L}{2}\right)\cdot \frac{3L}{2}$

$V = 10 \cdot 5 \cdot 15$

V = 750 units cubed

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ACT Math : Solid Geometry

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #972 : Act Math

Example Question #1 : How To Find The Length Of An Edge Of A Prism

Example Question #2 : Prisms

Example Question #2 : How To Find The Diagonal Of A Prism

Example Question #2 : Prisms

Example Question #977 : Act Math

Example Question #1 : How To Find The Surface Area Of A Prism

Example Question #1 : Non Cubic Prisms

Example Question #1 : How To Find The Surface Area Of A Prism

Example Question #1 : How To Find The Volume Of A Prism

All ACT Math Resources

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