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.

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1 answer

To find the missing length \( l \) of the rectangular prism, we can use the formula for the surface area of a rectangular prism, which is given by:

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

where \( l \) is the length, \( w \) is the width, and \( h \) is the height. In this case, we know:

  • Width \( w = 9 \) cm
  • Height \( h = 9 \) cm
  • Surface Area \( SA = 432 \) cm²

We can substitute the values into the surface area formula:

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

Simplifying the inside of the parentheses:

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

Next, we will isolate \( l \):

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

Thus, the missing length \( l \) of the rectangular prism is:

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