ACT Math : Solid Geometry

Study concepts, example questions & explanations for ACT Math

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Example Questions

Example Question #3 : How To Find The Length Of An Edge Of A Prism

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 \(\displaystyle 150\:m^3\). Sturgis knows that the exhibit will be \(\displaystyle 15\:m\) long and go \(\displaystyle 5\:m\) back into the wall.

What will the height of the new exhibit be?

Possible Answers:

\(\displaystyle 2\:m\)

\(\displaystyle 1\:m\)

\(\displaystyle 1.5\:m\)

\(\displaystyle 4\:m\)

\(\displaystyle 2.2\:m\)

Correct answer:

\(\displaystyle 2\:m\)

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:

\(\displaystyle Volume=length*width*height\)

Or

\(\displaystyle V=l*w*h\)

We are solving for height, so you can begin by rearranging the equation to get \(\displaystyle h\) by itself:

\(\displaystyle \frac{V}{l*w}=h\)

Then, plug in our knowns (\(\displaystyle V\), \(\displaystyle l\) and \(\displaystyle w\))

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

Here is the problem worked out with a corresponding picture:

Prism

Example Question #1 : Prisms

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 \(\displaystyle 150\:m^3\). Sturgis knows that the exhibit will be \(\displaystyle 15\:m\) long and go \(\displaystyle 5\:m\) back into the wall.

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

Possible Answers:

\(\displaystyle 112.5\:m\)

\(\displaystyle 1.0\:m\)

\(\displaystyle 1.25\:m\)

Cannot be determined with the information provided

\(\displaystyle 1.5\:m\)

Correct answer:

\(\displaystyle 1.5\:m\)

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:

\(\displaystyle 150\:m^3*\frac{3}{4}=112.5\:m^3\)

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

\(\displaystyle V=l*w*h\)

Becomes

\(\displaystyle \frac{V}{l*w}=h\)

\(\displaystyle h=\frac{112.5\:m^3}{15\:m*5\:m}=1.5\:m\)

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 \(\displaystyle h\):

\(\displaystyle \frac{V}{l*w}=h\)

At this point, we can substitute in our given values and solve for \(\displaystyle h\):

\(\displaystyle \frac{150\:m^3}{15\:m*5\:m}=2\:m\)

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

 \(\displaystyle h{}'=h*\frac{3}{4}\)

\(\displaystyle h{}'=2\:m*\frac{3}{4}=1.5\:m\)

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

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

Possible Answers:

\(\displaystyle \small 3\sqrt{13}\)

\(\displaystyle \small 2\sqrt{61}\)

\(\displaystyle \small 18\sqrt{2}\)

\(\displaystyle \small 13\sqrt{3}\)

\(\displaystyle \small 3\sqrt{29}\)

Correct answer:

\(\displaystyle \small 3\sqrt{29}\)

Explanation:

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

\(\displaystyle \small l^2+w^2=d^2\)

\(\displaystyle \small 6^2+9^2=d^2\)

\(\displaystyle \small 36+81=d^2\)

\(\displaystyle \small 117=d^2\)

\(\displaystyle \small d=\sqrt{117}\)

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

\(\displaystyle \small d^2+h^2=D^2\)

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

\(\displaystyle \small 117+144=D^2\)

\(\displaystyle \small D^2=261\)

\(\displaystyle \small D=\sqrt{261}=\sqrt{9}\sqrt{29}=3\sqrt{29}\)

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

Example Question #692 : Geometry

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

Possible Answers:

Cannot be determined

\(\displaystyle 5.7\)

\(\displaystyle 8.2\)

\(\displaystyle 7.2\)

\(\displaystyle 68\)

Correct answer:

\(\displaystyle 8.2\)

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 \(\displaystyle 4\sqrt2\)

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

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

\(\displaystyle a^2+b^2=c^2\)

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

\(\displaystyle 32+36=c^2\)

\(\displaystyle 68=c^2\)

\(\displaystyle \sqrt{68}=c\)

\(\displaystyle {\color{Blue} 8.2}=c\)

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

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

Possible Answers:

\(\displaystyle \sqrt{23}\)

\(\displaystyle 5\)

\(\displaystyle 5\sqrt2\)

\(\displaystyle 3\sqrt{2}\)

\(\displaystyle 4\sqrt2\)

Correct answer:

\(\displaystyle 5\sqrt2\)

Explanation:

Write the diagonal formula for a rectangular prism.

\(\displaystyle d=\sqrt{L^2+W^2+H^2}\)

Substitute and solve for the diagonal.

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

Example Question #66 : Solid Geometry

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:

\(\displaystyle \frac{1}{9}\)

\(\displaystyle \frac{\sqrt3}{3}\)

\(\displaystyle \sqrt3\)

\(\displaystyle \frac{\sqrt{1441}}{12}\)

\(\displaystyle \frac{\sqrt{1306}}{3}\)

Correct answer:

\(\displaystyle \frac{\sqrt{1441}}{12}\)

Explanation:

Convert the dimensions into feet.

\(\displaystyle 1 \textup{ yard}= 3\textup{ feet}\)

\(\displaystyle \textup{1 inch }= \frac{1}{12}\textup{ foot}\)

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

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

\(\displaystyle d=\sqrt{L^2+W^2+H^2}\)

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

\(\displaystyle d=\sqrt{\frac{1441}{144}} =\frac{\sqrt{1441}}{12}\textup{ ft}\)

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 \(\displaystyle 10\ ft \times 16\ ft\) carpet.  The ceiling is \(\displaystyle 8\ ft\) tall. He selects a paint that will cover \(\displaystyle 75\ ft^2\) per quart and \(\displaystyle 300\ ft^2\) per gallon. How much paint should he buy?

Possible Answers:

3 quarts

2 gallons and 1 quart

1 gallon and 1 quart

1 gallon and 2 quarts

1 gallon

Correct answer:

1 gallon and 2 quarts

Explanation:

Find the surface area of the walls: SAwalls = 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.

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

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

Possible Answers:

\(\displaystyle 130\ in^2\)

\(\displaystyle 100\ in^2\)

\(\displaystyle 120\ in^2\)

\(\displaystyle 140\ in^2\)

\(\displaystyle 25\ in^2\)

Correct answer:

\(\displaystyle 130\ in^2\)

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:

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

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

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

Substitute the values from the question to solve.

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

\(\displaystyle SA=50+40+40\)

\(\displaystyle SA=130\ in^2\)

Example Question #3 : 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:

\(\displaystyle 382\ in^2\)

None of the answers are correct  

\(\displaystyle 576\ in^2\)

\(\displaystyle 216\ in^2\)

\(\displaystyle 432\ in^2\)

Correct answer:

\(\displaystyle 432\ in^2\)

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 

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:

\(\displaystyle 250\) units cubed

\(\displaystyle 500\) units cubed

\(\displaystyle 300\) units cubed

\(\displaystyle 750\) units cubed

\(\displaystyle 150\) units cubed

Correct answer:

\(\displaystyle 750\) units cubed

Explanation:

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

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

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

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

\(\displaystyle V = 10 \cdot 5 \cdot 15\)

\(\displaystyle V = 750\) units cubed

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