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 a using Pythagorean theorem in base - triangle

The base of the prism is a right - triangle with sides 9 in and 12 in. By the Pythagorean theorem \(a=\sqrt{9^{2}+12^{2}}\).
\[a=\sqrt{81 + 144}=\sqrt{225}=15\]

Step2: Find b using Pythagorean theorem in 3 - D space

Now, consider a right - triangle in 3 - D space with sides a = 15 in and 7 in. By the Pythagorean theorem \(b=\sqrt{a^{2}+7^{2}}\). Substitute a = 15 into the formula: \(b=\sqrt{15^{2}+7^{2}}=\sqrt{225 + 49}=\sqrt{274}\approx16.6\)

Answer:

(a) 15
(b) 16.6