### All ACT Math Resources

## Example Questions

### Example Question #21 : Solid Geometry

Our backyard pool holds 10,000 gallons. Its average depth is 4 feet deep and it is 10 feet long. If there are 7.48 gallons in a cubic foot, how wide is the pool?

**Possible Answers:**

100 ft

33 ft

7.48 ft

30 ft

133 ft

**Correct answer:**

33 ft

There are 7.48 gallons in cubic foot. Set up a ratio:

1 ft^{3} / 7.48 gallons = x cubic feet / 10,000 gallons

Pool Volume = 10,000 gallons = 10,000 gallons * (1 ft^{3}/ 7.48 gallons) = 1336.9 ft^{3}

Pool Volume = 4ft x 10 ft x WIDTH = 1336.9 cubic feet

Solve for WIDTH:

4 ft x 10 ft x WIDTH = 1336.9 cubic feet

WIDTH = 1336.9 / (4 x 10) = 33.4 ft

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

A cube has a volume of 64cm^{3}. What is the area of one side of the cube?

**Possible Answers:**

4cm

16cm^{3}

4cm^{2}

16cm

16cm^{2}

**Correct answer:**

16cm^{2}

The cube has a volume of 64cm^{3}, making the length of one edge 4cm (4 * 4 * 4 = 64).

So the area of one side is 4 * 4 = 16cm^{2}

### Example Question #1 : Cubes

A cube as the volume of .

Find the length of a side of this cube.

**Possible Answers:**

**Correct answer:**

The formula to find the volume of the cube is

Since we know the volume, we can set up the equation

### Example Question #2 : Cubes

A cube has a surface area of , what is the length of the side of the cube? (If necessary, round to the nearest hundredth.)

**Possible Answers:**

**Correct answer:**

To find the length of the side of a square given the surface area, use the surface area formula and solve for :

, now divide both sides by 6

, now square root both sides

.

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

A certain cubic box when unfolded and laid flat on a table covers exactly square units of space. What is the width of the box, in units?

**Possible Answers:**

**Correct answer:**

To find the length of the edge of a cube from its surface area, remember that , where is the length of a side.

So, the box is units long.

### Example Question #4 : Cubes

Given the volume of a cube is , find the side length.

**Possible Answers:**

**Correct answer:**

To find side length, simply realize that volume is the side length cubed. Thus,

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

Find the length of the edge of a cube given the volume is .

**Possible Answers:**

**Correct answer:**

To solve, simply take the cube root of the volume. Thus,

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

Find the length of the diagonal of a cube with side length of .

**Possible Answers:**

**Correct answer:**

We begin with a picture, noting that the diagonal, labeled as , is the length across the cube from one vertex to the opposite side's vertex.

However, the trick to solving the problem is to also draw in the diagonal of the bottom face of the cube, which we labeled .

Note that this creates two right triangles. Though our end goal is to find , we can begin by looking at the right triangle in the bottom face to find . Using either the Pythagorean Theorem or the fact that we have a 45-45-90 right traingle, we can calculate the hypotenuse.

Now that we know the value of , we can turn to our second right triangle to find using the Pythagorean Theorem.

Taking the square root of both sides and simplifying gives the answer.

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

What is the diagonal length for a cube with volume of ? Round to the nearest hundredth.

**Possible Answers:**

**Correct answer:**

Recall that the volume of a cube is computed using the equation

, where is the length of one side of the cube.

So, for our data, we know:

Using your calculator, take the cube root of both sides. You can always do this by raising to the power if your calculator does not have a varied-root button.

If you get , the value really should be rounded up to . This is because of calculator estimations. So, if the sides are , you can find the diagonal by using a variation on the Pythagorean Theorem working for three dimensions:

This is . Round it to .

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

What is the length of the diagonal of a cube with a volume of ?

**Possible Answers:**

**Correct answer:**

Recall that the diagonal of a cube is most easily found when you know that cube's dimensions. For the volume of a cube, the pertinent equation is:

, where represents the length of one side of the cube. For our data, this gives us:

Now, you could factor this by hand or use your calculator. You will see that is .

Now, we find the diagonal by using a three-dimensional version of the Pythagorean Theorem / distance formula:

or

You can rewrite this:

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