### All ACT Math Resources

## Example Questions

### Example Question #55 : Solid Geometry

What is the sum of the number of vertices, edges, and faces of a cube?

**Possible Answers:**

None of the answers are correct

24

20

26

18

**Correct answer:**

26

Vertices = three planes coming together at a point = 8

Edges = two planes coming together to form a line = 12

Faces = one plane as the surface of the solid = 6

Vertices + Edges + Faces = 8 + 12 + 6 = 26

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

What is the difference in volume between a sphere with radius x and a cube with a side of 2x? Let π = 3.14

**Possible Answers:**

4.18x^{3 }

6.73x^{3 }

3.82x^{3}

8.00x^{3}

5.28x^{3 }

**Correct answer:**

3.82x^{3}

V_{cube} = s^{3} = (2x)^{3} = 8x^{3}

V_{sphere} = 4/3 πr^{3} = 4/3•3.14•x^{3} = 4.18x^{3 }

### Example Question #3 : How To Find The Volume Of A Figure

The density of gold is and the density of glass is . You have a gold cube that is in length on each side. If you want to make a glass cube that is the same weight as the gold cube, how long must each side of the glass cube be?

**Possible Answers:**

**Correct answer:**

Weight = Density * Volume

Volume of Gold Cube = side^{3}= x^{3}

Weight of Gold = 16 g/cm^{3 }* x^{3}

Weight of Glass = 3/cm^{3} * side^{3}

Set the weight of the gold equal to the weight of the glass and solve for the side length:

16* x^{3} = 2 * side^{3}

side^{3} = 16/2* x^{3 }= 8 x^{3}

Take the cube root of both sides:

side = 2x

### Example Question #3 : How To Find The Volume Of A Figure

Chemicals to clean a swimming pool cost $0.24 per cubic foot of water. If a pool is 6 feet deep, 14 feet long and 8 feet wide, how much does it cost to clean the pool? Round to the nearest dollar.

**Possible Answers:**

**Correct answer:**

The volume of the pool can be determined by multiplying the length, width, and height together.

Each cubit foot costs 24 cents, so:

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

A cube has edges that are three times as long as those of a smaller cube. The volume of the bigger cube is how many times larger than that of the smaller cube?

**Possible Answers:**

**Correct answer:**

If we let represent the length of an edge on the smaller cube, its volume is .

The larger cube has edges three times as long, so the length can be represented as . The volume is , which is .

The large cube's volume of is 27 times as large as the small cube's volume of .

### Example Question #2 : How To Find The Volume Of A Cube

A tank measuring 3in wide by 5in deep is 10in tall. If there are two cubes with 2in sides in the tank, how much water is needed to fill it?

**Possible Answers:**

**Correct answer:**

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

What is the volume of a cube that has a side length of inches?

**Possible Answers:**

**Correct answer:**

We are given the side length of a cube so we simply plug that into the formula for the volume of a cube.

That formula is , and so the correct answer is

.

Make sure to check your units, is the correct number, but the units are squared rather than cubed.

### Example Question #2 : How To Find The Volume Of A Cube

What is the volume of a cube with a side of length 1 cm?

**Possible Answers:**

**Correct answer:**

The formula for the volume of a cube with a side of length is:

### Example Question #3 : How To Find The Volume Of A Cube

A cube of sponge has volume . When water is added, the sponge triples in length along each dimension. What is the new volume of the cube, in cubic centimeters?

**Possible Answers:**

**Correct answer:**

If our original cube has a volume of , then the length of one of its edges is . Triple each edge to , then cube the result, and we obtain

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

Find the volume of a cube with side length 10.

**Possible Answers:**

**Correct answer:**

To find volume, simply cube the side length. Thus,