Question

Question content area top
Part 1
What is the longest line segment that can be drawn in a right rectangular prism that is 13 cm​ long, 8 cm​ wide, and 7 cm​ tall?

Answer 1

To find the longest line segment that can be drawn in a right rectangular prism (also known as a rectangular box), we need to calculate the length of the space diagonal. The formula to find the space diagonal \(d\) of a rectangular prism with length \(l\), width \(w\), and height \(h\) is given by:

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

In this case, we have:

- Length \(l = 13 \, \text{cm}\)
- Width \(w = 8 \, \text{cm}\)
- Height \(h = 7 \, \text{cm}\)

Now we can substitute these values into the formula:

\[
d = \sqrt{(13)^2 + (8)^2 + (7)^2}
\]

Calculating the squares:

\[
d = \sqrt{169 + 64 + 49}
\]

Adding these values together:

\[
d = \sqrt{282}
\]

Now calculating the square root:

\[
d \approx 16.82 \, \text{cm}
\]

Therefore, the longest line segment that can be drawn in the rectangular prism is approximately \(16.82 \, \text{cm}\).