Question

this prism has a lateral area of 560 square inches. the base is a right triangle. find the height of the prism. 17 in. 15 in. ? inches

Explanation:

Step1: Find the third - side of the base right - triangle

Using the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), for a right - triangle with legs \(a = 15\) in and \(b\) (unknown) and hypotenuse \(c = 17\) in. Let the other leg be \(x\), then \(x=\sqrt{17^{2}-15^{2}}=\sqrt{(17 + 15)(17 - 15)}=\sqrt{32\times2}=\sqrt{64}=8\) in.

Step2: Calculate the perimeter of the base

The perimeter \(P\) of the right - triangle base is \(P=8 + 15+17=40\) in.

Step3: Use the lateral - area formula

The lateral area \(L\) of a prism is given by \(L = Ph\), where \(P\) is the perimeter of the base and \(h\) is the height of the prism. We know \(L = 560\) square inches and \(P = 40\) in. Rearranging the formula for \(h\), we get \(h=\frac{L}{P}\).
Substitute \(L = 560\) and \(P = 40\) into the formula: \(h=\frac{560}{40}=14\) in.

Answer:

14