Question

What is the longest line segment that can be drawn in a right rectangular prism that is 14 cm​ long, 13 cm​ wide, and 10 cm​ tall?

Answer 1

To find the longest line segment that can be drawn in a right rectangular prism, you need to calculate the diagonal that spans from one vertex of the prism to the opposite vertex. The formula to find the length of the diagonal \(d\) of a rectangular prism with dimensions \(l\) (length), \(w\) (width), and \(h\) (height) is given by:

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

In this case, the dimensions of the prism are:

- Length \(l = 14 \, \text{cm}\)
- Width \(w = 13 \, \text{cm}\)
- Height \(h = 10 \, \text{cm}\)

Substituting these values into the formula gives:

\[
d = \sqrt{(14 \, \text{cm})^2 + (13 \, \text{cm})^2 + (10 \, \text{cm})^2}
\]

Calculating the squares:

\[
d = \sqrt{196 + 169 + 100}
\]

Now, add these values together:

\[
d = \sqrt{196 + 169 + 100} = \sqrt{465}
\]

Next, calculate \( \sqrt{465} \):

\[
d \approx 21.54 \, \text{cm}
\]

Thus, the longest line segment that can be drawn in the rectangular prism is approximately \( \text{21.54 cm} \).