Rectangular prism has edge lengths , , and , where is a positive constant. The surface area of prism is 208 cm. Rectangular prism is similar to prism and has surface area 468 cm. Which choice is the volume of prism , in cubic centimeters?

Solution available with step-by-step explanation

Super-Hard SAT Math Question 34 | Free SAT Prep | Aniko

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Question 34·200 Super-Hard SAT Math Questions·Geometry and Trigonometry

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Rectangular prism $A$ has edge lengths $x$ , $x+2$ , and $x+4$ , where $x$ is a positive constant. The surface area of prism $A$ is 208 cm $^2$ .

Rectangular prism $B$ is similar to prism $A$ and has surface area 468 cm $^2$ .

Which choice is the volume of prism $B$ , in cubic centimeters?

When similar solids are involved, separate the problem into two phases: (1) find the missing dimensions of the original solid using area or volume formulas, and (2) use similarity scaling rules ( $k^2$ for surface area, $k^3$ for volume). Always compute the linear scale factor from the surface-area ratio first, then cube it to scale volume.

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Set up surface area using three face pairs

Use $2(ab+ac+bc)$ with $a=x$ , $b=x+2$ , and $c=x+4$ , then set it equal to 208.

Find $x$ before thinking about prism $B$

Once you solve for $x$ , compute the volume of prism $A$ from $x(x+2)(x+4)$ .

Convert surface-area scaling to volume scaling

If $\dfrac{S_B}{S_A}=k^2$ for similar prisms, then $\dfrac{V_B}{V_A}=k^3$ .

Desmos Guide

Solve for $x$ from the surface area equation

Enter

$y=2(x(x+2)+x(x+4)+(x+2)(x+4))$

and also enter

$y=208$ .

Find the intersection and use the positive $x$ -value.

Compute the volume of prism $A$

In a new line, define $V_A = x(x+2)(x+4)$ .

Make sure Desmos is using the same $x$ you found from the intersection.

Scale the volume using the surface-area ratio

Compute $k = \sqrt{468/208}$ and then compute $V_B = V_A\cdot k^3$ .

Match the resulting value to the answer choices.

Step-by-step Explanation

Write and use the surface area formula for prism $A$

For a rectangular prism with side lengths $a$ , $b$ , and $c$ , the surface area is $2(ab+ac+bc)$ .

Here, $a=x$ , $b=x+2$ , and $c=x+4$ , so

\begin{aligned} 208 &= 2\big(x(x+2)+x(x+4)+(x+2)(x+4)\big) \\ &= 2\big((x^2+2x)+(x^2+4x)+(x^2+6x+8)\big) \\ &= 2(3x^2+12x+8). \end{aligned}

Divide by 2:

3x^2+12x+8=104.

Solve for $x$ and find the volume of prism $A$

Continue solving:

\begin{aligned} 3x^2+12x+8 &= 104 \\ 3x^2+12x-96 &= 0 \\ x^2+4x-32 &= 0. \end{aligned}

Factor:

(x+8)(x-4)=0.

Since $x$ is positive, $x=4$ . So prism $A$ has side lengths 4, 6, and 8, and its volume is

V_A = 4\cdot 6\cdot 8 = 192\text{ cm}^3.

Use similarity to scale from surface area to volume

For similar solids, surface areas scale by the square of the linear scale factor $k$ .

\frac{S_B}{S_A} = k^2 = \frac{468}{208} = \frac{9}{4} \quad\Rightarrow\quad k = \frac{3}{2}.

Volumes scale by $k^3$ , so

V_B = V_A\cdot k^3 = 192\left(\frac{3}{2}\right)^3 = 192\cdot \frac{27}{8} = 648.

Therefore, the volume of prism $B$ is $648$ cubic centimeters.

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation