Question

what is the missing length of a rectangular prism if the height and width are both 8 inches and the surface area is 256 in2. use proper units

Answer 1

To find the missing length of a rectangular prism when the height and width are known, we'll use the formula for the surface area of a rectangular prism:

\[
SA = 2(lw + lh + wh)
\]

where:
- \( SA \) is the surface area,
- \( l \) is the length,
- \( w \) is the width,
- \( h \) is the height.

Given:
- \( w = 8 \) inches (width)
- \( h = 8 \) inches (height)
- \( SA = 256 \) in² (surface area)

We can substitute these values into the surface area formula:

\[
256 = 2(l \cdot 8 + l \cdot 8 + 8 \cdot 8)
\]

Simplifying the equation:

\[
256 = 2(8l + 8l + 64)
\]

\[
256 = 2(16l + 64)
\]

\[
256 = 32l + 128
\]

Next, we can isolate \( l \) by first subtracting 128 from both sides:

\[
256 - 128 = 32l
\]

\[
128 = 32l
\]

Now, divide both sides by 32:

\[
l = \frac{128}{32} = 4 \text{ inches}
\]

Thus, the missing length of the rectangular prism is:

\[
\boxed{4 \text{ inches}}
\]