Question

type the correct answer in the box. use numerals instead of words.
the composite figure shown has a surface area of 482 square inches
what is the length of the prism in inches?
the length of the prism is inches.

Explanation:

Step1: Identify surface - area formula for composite figure

The composite figure can be thought of in terms of rectangular - shaped faces. The surface area formula for a rectangular prism - like composite figure is \(SA = 2(lw+lh + wh)\) for the main part and additional areas from the slanted part. Let the length of the prism be \(x\) inches.
The surface area of the figure is composed of the following faces:
Two faces with dimensions \(x\times3\), two faces with dimensions \(x\times(5 + 4)\), two faces with dimensions \(3\times4\), two faces with dimensions \(3\times5\).
The surface - area formula \(SA=2(3x)+2(9x)+2(3\times4)+2(3\times5)\).

Step2: Simplify the surface - area formula

\[

$$\begin{align*} SA&=6x + 18x+24 + 30\\ SA&=24x+54 \end{align*}$$

\]

Step3: Solve for \(x\)

We know that \(SA = 482\) square inches. So we set up the equation \(24x+54 = 482\).
Subtract 54 from both sides: \(24x=482 - 54=428\).
Then divide both sides by 24: \(x=\frac{428}{24}=\frac{107}{6}\approx17.83\).

Answer:

\(\frac{107}{6}\)