Question

the right isosceles triangle shown is rotated about line k with the base forming perpendicular to k. the perimeter of the triangle is 58 units. which best describes the resulting three - dimensional shape? a cone with a base radius of 17 units a cone with a base radius of 34 units a cylinder with a base radius of 17 units a cylinder with a base radius of 34 units

Explanation:

Step1: Recall 3 - D shape formed by rotation

When a right - triangle is rotated about an axis perpendicular to its base, the resulting three - dimensional shape is a cone.

Step2: Find the base radius of the triangle

Let the base and height of the right - isosceles triangle be \(x\) and the hypotenuse be \(h\). In a right - isosceles triangle, if the legs are of length \(x\), then \(h=\sqrt{x^{2}+x^{2}}=\sqrt{2x^{2}}=\sqrt{2}x\). The perimeter \(P = 2x + h=2x+\sqrt{2}x=(2 + \sqrt{2})x\). Given \(P = 58\), so \(x=\frac{58}{2+\sqrt{2}}=\frac{58(2 - \sqrt{2})}{(2+\sqrt{2})(2 - \sqrt{2})}=\frac{116-58\sqrt{2}}{4 - 2}=\frac{116-58\sqrt{2}}{2}=58 - 29\sqrt{2}\approx58-29\times1.414 = 58 - 40.906=17\) (approx). The base radius of the cone formed by rotation is the length of the leg of the right - triangle, which is approximately 17 units.

Answer:

A. a cone with a base radius of 17 units