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)

1 answer

To find the missing length \( l \) of a rectangular prism when the height \( h \) and width \( w \) are both given, and the surface area \( SA \) is known, we start with the formula for the surface area of a rectangular prism:

\[ SA = 2(hw + hl + wl) \]

where:

  • \( h \) is the height,
  • \( w \) is the width,
  • \( l \) is the length.

In this problem, we know:

  • \( h = 9 \) cm,
  • \( w = 9 \) cm,
  • \( SA = 432 \) cm\(^2\).

Substituting the known values into the surface area formula:

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

This simplifies to:

\[ 432 = 2(81 + 9l + 9l) \] \[ 432 = 2(81 + 18l) \] \[ 432 = 162 + 36l \]

Now, isolate \( l \):

\[ 432 - 162 = 36l \] \[ 270 = 36l \] \[ l = \frac{270}{36} \] \[ l = 7.5 \]

Thus, the missing length \( l \) of the rectangular prism, rounded to the nearest tenth, is:

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