Question

What is the length of a box with a surface area of 472 square cm and a width of 11 cm and height of 6 cm?

Answer 1

The surface area \( S \) of a rectangular box (also known as a rectangular prism) is given by the formula:

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

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

Given:
- Surface area \( S = 472 \) square cm,
- Width \( w = 11 \) cm,
- Height \( h = 6 \) cm.

We can substitute the values of \( w \) and \( h \) into the surface area formula to find the length \( l \).

First, let's substitute \( w \) and \( h \):

\[
472 = 2(l \cdot 11 + l \cdot 6 + 11 \cdot 6)
\]

Now, calculate \( 11 \cdot 6 \):

\[
11 \cdot 6 = 66
\]

Substitute back into the equation:

\[
472 = 2(l \cdot 11 + l \cdot 6 + 66)
\]

Divide both sides by 2:

\[
236 = l \cdot 11 + l \cdot 6 + 66
\]

Combine like terms:

\[
236 = l(11 + 6) + 66
\]

\[
236 = l \cdot 17 + 66
\]

Now subtract 66 from both sides:

\[
236 - 66 = l \cdot 17
\]

\[
170 = l \cdot 17
\]

Now, solve for \( l \):

\[
l = \frac{170}{17} = 10
\]

Thus, the length of the box is \( \boxed{10} \) cm.