Question

What is the missing length of a rectangular prism where the height and width are both 9 cm and the surface area is 432 cm squared? Express answer as a decimal rounded to the nearest tenth.

Answer 1

To find the missing length of a rectangular prism given the height and width, and the surface area, we start with the formula for the surface area \( S \) of a rectangular prism:

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

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

In this problem, we are given:
- \( h = 9 \) cm,
- \( w = 9 \) cm,
- \( S = 432 \) cm².

We need to find the length \( l \). Substituting the known values into the surface area formula gives:

\[
432 = 2(l \cdot 9 + l \cdot 9 + 9 \cdot 9)
\]

Simplifying inside the parentheses:

\[
432 = 2(9l + 9l + 81)
\]

This simplifies further to:

\[
432 = 2(18l + 81)
\]

Dividing both sides by 2:

\[
216 = 18l + 81
\]

Subtracting 81 from both sides:

\[
216 - 81 = 18l
\]
\[
135 = 18l
\]

Now, we solve for \( l \):

\[
l = \frac{135}{18}
\]

Calculating the division:

\[
l = 7.5
\]

Thus, the missing length of the rectangular prism is:

\[
\boxed{7.5} \text{ cm}
\]