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 cm2
? Express your answer as a decimal rounded to the nearest tenth.(1 point)
cm

Answer 1

To find the missing length (depth) \( l \) of a rectangular prism given the height \( h = 9 \) cm, width \( w = 9 \) cm, and surface area \( SA = 432 \) cm², we can use the formula for the surface area of a rectangular prism:

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

Substituting the known values:

1. \( SA = 432 \)
2. \( h = 9 \)
3. \( w = 9 \)

Thus, the equation becomes:

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

This simplifies to:

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

Combine like terms:

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

Now distribute the 2:

\[
432 = 36l + 162
\]

Next, isolate \( l \) by subtracting 162 from both sides:

\[
432 - 162 = 36l
\]

\[
270 = 36l
\]

Now, divide both sides by 36 to solve for \( l \):

\[
l = \frac{270}{36} = 7.5
\]

Rounded to the nearest tenth, the missing length is:

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