Prisms

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SAT Math › Prisms

Questions 1 - 10

A rectangular prism has a length that is twice as long as its width, and a width that is twice as long as its height. If the surface area of the prism is 252 square units, what is the volume, in cubic units, of the prism?

108

432

216

1728

Explanation

Let l be the length, w be the width, and h be the height of the prism. We are told that the length is twice the width, and that the width is twice the height. We can set up the following two equations:

l = 2_w_

w = 2_h_

Next, we are told that the surface area is equal to 252 square units. Using the formula for the surface area of the rectangular prism, we can write the following equation:

surface area = 2_lw_ + 2_lh_ + 2_wh_ = 252

We now have three equations and three unknowns. In order to solve for one of the variables, let's try to write w and l in terms of h. We know that w = 2_h_. Because l = 2_w_, we can write l as follows:

l = 2_w_ = 2(2_h_) = 4_h_

Now, let's substitute w = 2_h_ and l = 4_h_ into the equation we wrote for surface area.

2(4_h_)(2_h_) + 2(4_h_)(h) + 2(2_h_)(h) = 252

Simplify each term.

16_h_2 + 8_h_2 + 4_h_2 = 252

Combine _h_2 terms.

28_h_2 = 252

Divide both sides by 28.

_h_2 = 9

Take the square root of both sides.

h = 3.

This means that h = 3. Because w = 2_h_, the width must be 6. And because l = 2_w_, the length must be 12.

Because we now know the length, width, and height, we can find the volume of the prism, which is what the question ultimately requires us to find.

volume of a prism = l • w • h

volume = 12(6)(3)

= 216 cubic units

The answer is 216.

Box

The shaded face of the above rectangular prism is a square. In terms of , give the surface area of the prism.

$2 x^{2} + 100x$

$4 x^{2} + 50x$

$4 x^{2} + 1,250$

Explanation

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

There are two faces (front and back) that are squares of sidelength ; the area of each is the square of this, or $x^{2}$ .

There are four faces (left, right, top, bottom) that are rectangles of dimensions 25 and ; the area of each is the product of the two, .

The surface area is the total of their areas:

$2 \cdot x^{2} + 4 \cdot 25x$

$= 2 x^{2} + 100x$

108

432

216

1728

Explanation

l = 2_w_

w = 2_h_

Next, we are told that the surface area is equal to 252 square units. Using the formula for the surface area of the rectangular prism, we can write the following equation:

surface area = 2_lw_ + 2_lh_ + 2_wh_ = 252

l = 2_w_ = 2(2_h_) = 4_h_

Now, let's substitute w = 2_h_ and l = 4_h_ into the equation we wrote for surface area.

2(4_h_)(2_h_) + 2(4_h_)(h) + 2(2_h_)(h) = 252

Simplify each term.

16_h_2 + 8_h_2 + 4_h_2 = 252

Combine _h_2 terms.

28_h_2 = 252

Divide both sides by 28.

_h_2 = 9

Take the square root of both sides.

h = 3.

This means that h = 3. Because w = 2_h_, the width must be 6. And because l = 2_w_, the length must be 12.

Because we now know the length, width, and height, we can find the volume of the prism, which is what the question ultimately requires us to find.

volume of a prism = l • w • h

volume = 12(6)(3)

= 216 cubic units

The answer is 216.

A given cube with a volume of is split into two equal prisms. What is the volume of one of the prisms?

$\frac{30}{\sqrt{2}}$

$\frac{30}{7}$

$\frac{20}{\sqrt{5}}$

Explanation

The question tells us that the cube is split into two prisms of equal size. Therefore, we don't even need to use the formula for the volume of a prism. If the cube is split into two equal forms, then the two prisms will be equal to one-half of the cube's volume.

$40 \div 2 = 20$

Box

The shaded face of the above rectangular prism is a square. In terms of , give the surface area of the prism.

$2 x^{2} + 100x$

$4 x^{2} + 50x$

$4 x^{2} + 1,250$

Explanation

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

There are two faces (front and back) that are squares of sidelength ; the area of each is the square of this, or $x^{2}$ .

There are four faces (left, right, top, bottom) that are rectangles of dimensions 25 and ; the area of each is the product of the two, .

The surface area is the total of their areas:

$2 \cdot x^{2} + 4 \cdot 25x$

$= 2 x^{2} + 100x$

A given cube with a volume of is split into two equal prisms. What is the volume of one of the prisms?

$\frac{30}{\sqrt{2}}$

$\frac{30}{7}$

$\frac{20}{\sqrt{5}}$

Explanation

$40 \div 2 = 20$

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?

$4\sqrt{3}$

$4\sqrt{2}$

Explanation

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.

$D=\sqrt{65}$

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.

$L^2=(4)^2+(\sqrt{65})^2$

$L=\sqrt{81}=9$

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.

$111 \textrm{ cm}$

$313 \textrm{ cm}$

$160 \textrm{ cm}$

$144\textrm{ cm}$

The correct answer is not among the other responses.

Explanation

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

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

$L = \frac{2}{3} W$

$\frac{3} {2} \cdot L = \frac{3} {2} \cdot \frac{2}{3} W$

$\frac{3} {2} L = W$

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

$L = \frac{3}{4} H$

$\frac{4} {3} \cdot L = \frac{4} {3} \cdot \frac{3}{4} H$

$\frac{4} {3} L = H$

The dimensions of the crate in terms of are , $\frac{3} {2} L$ , and $\frac{4} {3} L$ . The surface area is found using the formula:

Substitute:

$2\left ( L \cdot \frac{3} {2} L + L \cdot \frac{4} {3} L + \frac{3} {2} L \cdot \frac{4} {3} L \right ) =12$

$2\left ( \frac{3} {2} L^{2} + \frac{4} {3} L^{2} +2 L^{2} \right ) =12$

$2\left ( \frac{29} {6} L^{2} \right ) =12$

$\frac{29} {3} L^{2} =12$

Solve for :

$\frac{3} {29} \cdot \frac{29} {3} L^{2} =\frac{3} {29} \cdot 12$

$L^{2} =\frac{36} {29}$

$L =\sqrt{\frac{36} {29}}$

$L \approx 1.114$ meters.

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

$1.114 \times 100 = 111.4$

This rounds to 111 centimeters.

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?

$4\sqrt{3}$

$4\sqrt{2}$

Explanation

$D=\sqrt{65}$

$L^2=(4)^2+(\sqrt{65})^2$

$L=\sqrt{81}=9$

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.

$111 \textrm{ cm}$

$313 \textrm{ cm}$

$160 \textrm{ cm}$

$144\textrm{ cm}$

The correct answer is not among the other responses.

Explanation

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

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

$L = \frac{2}{3} W$

$\frac{3} {2} \cdot L = \frac{3} {2} \cdot \frac{2}{3} W$

$\frac{3} {2} L = W$

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

$L = \frac{3}{4} H$

$\frac{4} {3} \cdot L = \frac{4} {3} \cdot \frac{3}{4} H$

$\frac{4} {3} L = H$

The dimensions of the crate in terms of are , $\frac{3} {2} L$ , and $\frac{4} {3} L$ . The surface area is found using the formula:

Substitute:

$2\left ( L \cdot \frac{3} {2} L + L \cdot \frac{4} {3} L + \frac{3} {2} L \cdot \frac{4} {3} L \right ) =12$

$2\left ( \frac{3} {2} L^{2} + \frac{4} {3} L^{2} +2 L^{2} \right ) =12$

$2\left ( \frac{29} {6} L^{2} \right ) =12$

$\frac{29} {3} L^{2} =12$

Solve for :

$\frac{3} {29} \cdot \frac{29} {3} L^{2} =\frac{3} {29} \cdot 12$

$L^{2} =\frac{36} {29}$

$L =\sqrt{\frac{36} {29}}$

$L \approx 1.114$ meters.

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

$1.114 \times 100 = 111.4$

This rounds to 111 centimeters.

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