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=square ) 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=square ) in

Explanation:

Step1: Find \(a\) using the Pythagorean theorem for the base - related right triangle

The base of the right - triangular prism has a right triangle with legs \(6\) in and \(8\) in. By the Pythagorean theorem \(c^{2}=x^{2}+y^{2}\) (where \(c = a\), \(x = 6\), \(y = 8\)).

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

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

$$a^{2}=100$$

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

Step2: Find \(b\) using the Pythagorean theorem for the larger right triangle

Now, consider the right triangle with legs \(a = 10\) in and \(12\) in. Let the hypotenuse be \(b\). By 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