### All SAT Math Resources

## Example Questions

### Example Question #1 : Prisms

For a box to fit inside the cupboard, the sum of the height and the perimeter of the box must, at most, be 360 cm. If Jenn has a box that has a height of 40 cm and a length of 23 cm, what is the greatest possible width of the box?

**Possible Answers:**

207 cm

13 cm

0.4 cm

297 cm

137 cm

**Correct answer:**

137 cm

First we write out the equation we are given. *H* + (2*L* +2*W*) = 360. *H *= 40 and *L *= 23

40 + (2(23) + 2*W*) = 360

40 + (46 + 2*W*) = 360

46 + 2*W* = 320

2*W* = 274

*W* = 137

### Example Question #2 : Prisms

The volume of a rectangular prism is 80 cm^{3}. The length, width, and height of the prism are each an integer number of cm. If the dimensions form three terms of an arithmetic sequence, find the average of the three dimensions.

**Possible Answers:**

5

8

6

4

7

**Correct answer:**

5

Method 1:

Trial and error to find a combination of factors of 80 that differ by the same amount will eventually yield 2, 5, 8. The average is 5.

Method 2:

Three terms of an arithmetic sequence can be written as x, x+d, and x+2d. Multiply these together using the distributive property to find the volume and the following equation results:

x^{3} + 3dx^{2} + 2d^{2}x - 80 = 0

Find an integer value of x that creates an integer solution for d. Try x=1 and we see the equation 1 + 3d + 2d^{2} - 80 = 0 or 2d^{2} + 3d -79 = 0. The determinant of this quadratic is 641, which is not a perfect square. Therefore, d is not an integer when x=1.

Try x=2 and we see the equation 8 + 12d + 4d^{2} - 80 = 0 or d^{2} + 3d - 18 = 0. This is easily factored to (d+6)(d-3)=0 so d=-6 or d=3. Since a negative value of d will result in negative dimensions of the prism, d must equal 3. Therefore, when substituting x=2 and d=3, the dimensions x, x+d, and x+2d become 2, 5, and 8. The average is 5.

### Example Question #3 : Prisms

A right rectangular prism has a volume of 64 cubic units. Its dimensions are such that the second dimension is twice the length of the first, and the third is one-fourth the dimension of the second. What are its exact dimensions?

**Possible Answers:**

3 x 6 x 12

4 x 4 x 4

1 x 2 x 32

1 x 4 x 16

2 x 4 x 8

**Correct answer:**

2 x 4 x 8

Based on our prompt, we can say that the prism has dimensions that can be represented as:

Dim1: x

Dim2: 2 * Dim1 = 2x

Dim3: (1/4) * Dim2 = (1/4) * 2x = (1/2) * x

More directly stated, therefore, our dimensions are: x, 2x, and 0.5x. Therefore, the volume is x * 2x * 0.5x = 64, which simplifies to x^{3} = 64. Solving for x, we find x = 4. Therefore, our dimensions are:

x = 4

2x = 8

0.5x = 2

Or: 2 x 4 x 8

### Example Question #4 : Prisms

A right rectangular prism has a volume of 120 cubic units. Its dimensions are such that the second dimension is three times the length of the first, and the third dimension is five times the dimension of the first. What are its exact dimensions?

**Possible Answers:**

1 x 5 x 24

2 x 6 x 10

4 x 12 x 20

2 x 5 x 12

None of the other answers

**Correct answer:**

2 x 6 x 10

Based on our prompt, we can say that the prism has dimensions that can be represented as:

Dim1: x

Dim2: 3 * Dim1 = 3x

Dim3: 5 * Dim1 = 5x

More directly stated, therefore, our dimensions are: x, 3x, and 5x. Therefore, the volume is x * 3x * 5x = 120, which simplifies to 15x^{3} = 120 or x^{3} = 8. Solving for x, we find x = 2. Therefore, our dimensions are:

x = 2

3x = 6

5x = 10

Or: 2 x 6 x 10

### Example Question #5 : Prisms

The length of a crate is three-fourths its height and two-thirds its width. The surface area of the crate is 12 square meters. To the nearest *centimeter*, give the length of the box.

**Possible Answers:**

The correct answer is not among the other responses.

**Correct answer:**

Call , , and the length, height, and width of the crate.

The length of the crate is two-thirds its width, so

The length of the crate is three-fourths its height, so

The dimensions of the crate in terms of are , , and . The surface area is found using the formula:

Substitute:

Solve for :

meters.

Since one meter comprises 100 centimeters, multiply by 100 to convert to centimeters:

This rounds to 111 centimeters.

### Example Question #6 : Prisms

The shaded face of the above rectangular prism is a square. If the volume of the prism is , give in terms of .

**Possible Answers:**

**Correct answer:**

Since the top face of the prism is a square, the common sidelength - and the missing dimension - is 25.

The volume of a rectangular prism is equal to the product of its length, width, and height - that is,

Setting , and solving for :

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

The dimensions of a right, rectangular prism are 4 in x 12 in x 2 ft. What is the diagonal distance of the prism?

**Possible Answers:**

4√(3)

8√(7)

4√(46)

8√(23)

None of the other answers

**Correct answer:**

4√(46)

The problem is simple, but **be careful**. **The units are not equal. **First convert the last dimension into inches. There are 12 inches per foot. Therefore, the prism's dimensions really are: 4 in x 12 in x 24 in.

From this point, things are relatively easy. The distance from corner to corner in a three-dimensional prism like this can be found by using a variation on the Pythagorean Theorem that merely adds one dimension. That is, *d*^{2} = *x*^{2} + *y*^{2} + *z*^{2}, or *d* = √(*x*^{2} + *y*^{2} + *z*^{2})

For our data, this would be:

*d* = √(4^{2} + 12^{2} + 24^{2}) = √(16 + 144 + 576) = √(736) = √(2 * 2 * 2 * 2 * 2 * 23) = 4√(46)

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

The base of a right, rectangular prism is a square. Its height is three times that of one of the sides of the base. If its overall volume is 375 in^{3}, what is the diagonal distance of the prism?

**Possible Answers:**

25√(11) in

5 in

5√(11) in

5√(3) in

None of the other answers

**Correct answer:**

5√(11) in

First, let's represent our dimensions. We know the bottom could be represented as being *x* by *x*. The height is said to be three times one of these dimensions, so let's call it 3*x*. Based on this, we know the dimensions of the prism are *x*, *x*, and 3*x*. Now, the volume of a right rectangular prism is found by multiplying together its three dimensions. Therefore, if we know the overall volume is 375 in^{3}, we can say:

375 = *x* * *x* * 3*x* or 375 = 3*x*^{3}

Simplifying, we first divide by 3: 125 = *x*^{3}. Taking the cube root of both sides, we find that *x* = 5.

**Now, be careful**. The dimensions **are not** 5, 5, 5. They are (recall) *x*, *x*, and 3*x*. If *x* = 5, this means the dimensions are 5, 5, and 15.

At this point, things are beginning to progress to the end of the problem. The distance from corner to corner in a three-dimensional prism like this can be found by using a variation on the Pythagorean Theorem that merely adds one dimension. That is, *d*^{2} = *x*^{2} + *y*^{2} + *z*^{2}, or *d* = √(*x*^{2} + *y*^{2} + *z*^{2})

For our data, this would be: *d* = √(5^{2} + 5^{2} + 15^{2}) = √(25 + 25 + 225) = √(275) = √(5 * 5 * 11) = 5√(11) in

### Example Question #3 : How To Find The Diagonal Of A Prism

The base of a right, rectangular prism has one side that is three times the length of the other. Its height is twice the length of the longer side of the base. If its overall volume is 13,122 in^{3}, what is the diagonal distance of the prism?

**Possible Answers:**

9√(23) in

9√(46) in

None of the other answers

6√(23) in

9√(13) in

**Correct answer:**

9√(46) in

First, let's represent our dimensions. We know the bottom could be represented as being *x* by 3*x*. The height is said to be twice the longer dimension, so let's call it 2 * 3*x*, or 6*x*. Based on this, we know the dimensions of the prism are *x*, 2*x*, and 6*x*. Now, the volume of a right rectangular prism is found by multiplying together its three dimensions. Therefore, if we know the overall volume is 13,122 in^{3}, we can say:

13,122 = *x* * 3*x* * 6*x* or 13,122 = 18*x*^{3}

Simplifying, we first divide by 18: 729 = *x*^{3}. Taking the cube root of both sides, we find that *x* = 9.

**Now, be careful**. The dimensions **are not** 9, 9, and 9. They are (recall) *x*, 3*x*, and 6*x*. If *x* = 9, this means the dimensions are 9, 27, and 54.

At this point, things are beginning to progress to the end of the problem. The distance from corner to corner in a three-dimensional prism like this can be found by using a variation on the Pythagorean Theorem that merely adds one dimension. That is, *d*^{2} = *x*^{2} + *y*^{2} + *z*^{2}, or *d* = √(*x*^{2} + *y*^{2} + *z*^{2})

For our data, this would be: *d* = √(9^{2} + 27^{2} + 54^{2}) = √(81 + 729 + 2916) = √(3726) = √(2 * 3 * 3 * 3 * 3 * 23) = 9√(46) in.

### Example Question #4 : How To Find The Diagonal Of A Prism

A rectangular prism has length 7, width 4, and height 4. What is the distance from the top back left corner to the bottom front right corner?

**Possible Answers:**

**Correct answer:**

The diagonal from the top back left corner to the bottom front right corner will be the hypotenuse of a right triangle. The sides of the triangle will be the height of the box and the diagonal through the middle of one of the rectangular faces. We will be able to solve for the length using the Pythagorean Theorem.

To calculate the length of the hypotenuse, we first must find the length of the rectangular diagonal using the sides of the rectangle. This diagonal will be the hypotenuse of a right triangle with sides 7 and 4. Solve for the diagonal length using the Pythagorean Theorem.

Now we can return to our first triangle. We are given the height, 4, and now have the length of the rectangular diagonal. Use these values to solve for the length of the diagonal that connects the top back left corner and the bottom front right corner.

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