Question

a gift box has the shape of a right triangular prism. a piece of tape of length b (in inches) is placed from one corner to the opposite corner, as shown in the figure. (the figure is not drawn to scale.) (a) find a. a = in (b) use your answer to part (a) to find b, the length of the tape. round your answer to the nearest tenth of an inch. b = in

Explanation:

Step1: Find the length of the base - diagonal (a) using Pythagorean theorem

In the base right - triangle with sides 9 in and 12 in, by the Pythagorean theorem \(a^{2}=9^{2}+12^{2}\).
\[

$$\begin{align*} a^{2}&=81 + 144\\ a^{2}&=225\\ a&=\sqrt{225}\\ a& = 15 \end{align*}$$

\]

Step2: Find the length of the tape (b) using Pythagorean theorem again

Now, consider the right - triangle with sides \(a = 15\) in and 5 in. By the Pythagorean theorem \(b^{2}=a^{2}+5^{2}\). Since \(a = 15\), then \(b^{2}=15^{2}+5^{2}\).
\[

$$\begin{align*} b^{2}&=225+25\\ b^{2}&=250\\ b&=\sqrt{250}\\ b&\approx15.8 \end{align*}$$

\]

Answer:

a = 15 in
b = 15.8 in