Question

A box shaped like a right rectangular prism measures 8 inches by 6 inches by 5 inches. What is the length of the interior diagonal of the prism to the nearest hundredth?
10.77 inches
11.18 inches
10.68 inches
11.92 inches

Answer 1

To find the length of the interior diagonal of a right rectangular prism, we can use the formula for the diagonal \( d \) of a prism:

\[
d = \sqrt{l^2 + w^2 + h^2}
\]

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

In this case, the dimensions of the box are:
- \( l = 8 \) inches,
- \( w = 6 \) inches, and
- \( h = 5 \) inches.

Now we can substitute these values into the formula:

\[
d = \sqrt{8^2 + 6^2 + 5^2}
\]

Calculating each term:

\[
8^2 = 64
\]
\[
6^2 = 36
\]
\[
5^2 = 25
\]

Now, add these values together:

\[
64 + 36 + 25 = 125
\]

Next, we take the square root of 125:

\[
d = \sqrt{125}
\]

We can simplify \( \sqrt{125} \):

\[
\sqrt{125} = \sqrt{25 \cdot 5} = 5\sqrt{5}
\]

Now we need to calculate \( 5\sqrt{5} \). The approximate value of \( \sqrt{5} \) is about \( 2.236 \):

\[
d \approx 5 \times 2.236 \approx 11.18
\]

Thus, the length of the interior diagonal of the prism, to the nearest hundredth, is:

\[
\boxed{11.18} \text{ inches}
\]