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 base - diagonal length \(a\)

In the base right - triangle with sides 6 in and 8 in, use the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\). Here, \(a = \sqrt{6^{2}+8^{2}}\).
\[a=\sqrt{36 + 64}=\sqrt{100}=10\]

Step2: Find the length \(b\)

Now, consider the right - triangle formed by \(a = 10\) in, height 5 in and the diagonal \(b\). Again, use the Pythagorean theorem \(b=\sqrt{a^{2}+h^{2}}\), where \(a = 10\) and \(h = 5\).
\[b=\sqrt{10^{2}+5^{2}}=\sqrt{100 + 25}=\sqrt{125}\approx11.2\]

Answer:

(a) \(a = 10\) in
(b) \(b\approx11.2\) in