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 = 10 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 = 15.6 in

Explanation:

Step1: Find the length of the base diagonal

We can use the Pythagorean theorem for the right - angled triangle with legs \(6\) in and \(8\) in.
The Pythagorean theorem is \(c^{2}=x^{2}+y^{2}\), where \(x = 6\), \(y = 8\) and \(c=a\).

$$a^{2}=6^{2}+8^{2}$$

$$a^{2}=36 + 64$$

$$a^{2}=100$$

$$a=\sqrt{100}=10$$

Step2: Find the length of the tape \(b\)

Now, consider the right - angled triangle with legs \(a = 10\) in and \(12\) in. Let the hypotenuse be \(b\).
Using the Pythagorean theorem \(b^{2}=a^{2}+12^{2}\)
Substitute \(a = 10\) into the formula:

$$b^{2}=10^{2}+12^{2}$$

$$b^{2}=100+144$$

$$b^{2}=244$$

$$b=\sqrt{244}\approx15.6$$

Answer:

(a) \(a = 10\) in
(b) \(b\approx15.6\) in