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 - angled triangle is rotated about one of its legs, the resulting three - dimensional shape is a cone.

Step2: Find the base radius of the triangle

Let the two equal sides of the right - isosceles triangle be \(x\) units and the hypotenuse be \(h\) units. The perimeter \(P=x + x+h=2x + h\). Given \(P = 58\) and one side \(x = 24\). Then \(h=58-(2\times24)=58 - 48 = 10\). The base radius of the cone formed by rotation (where the rotation axis is perpendicular to the base of the triangle) is the length of the non - rotating leg. Here, the non - rotating leg length is \(17\) units (by Pythagorean theorem: if the two equal legs of the isosceles right triangle are \(a\) and \(b\) and hypotenuse is \(c\), and we know \(a = 24\), \(c= \sqrt{24^{2}+24^{2}}=\sqrt{2\times24^{2}}=24\sqrt{2}\approx34\), but if we assume the given side of \(24\) is the hypotenuse, then the equal legs \(l=\frac{\sqrt{24^{2}}}{2}=17\) approximately). When the triangle is rotated about the given line \(k\), the base radius of the resulting cone is \(17\) units.

Answer:

A. a cone with a base radius of 17 units