Question

identifying cross sections and finding area
what is the type of two - dimensional solid created by a vertical cross section of the cone that passes through the apex? what is the area of the cross section?
○ triangle; area = 45 ft²
○ triangle; area = 90 ft²
○ circle; area = 36π ft²
○ circle; area = 144π ft²

Explanation:

Step1: Determine the cross - section shape

A vertical cross - section of a cone that passes through the apex (the tip of the cone) and is vertical will form a triangle. The base of the triangle is the diameter of the base of the cone, and the height of the triangle is the height of the cone. The radius of the base of the cone is \(r = 6\) ft, so the diameter \(d=2r = 12\) ft. The height of the cone \(h = 15\) ft.

Step2: Calculate the area of the triangular cross - section

The formula for the area of a triangle is \(A=\frac{1}{2}\times base\times height\). Here, the base of the triangle (diameter of the cone's base) is \(b = 2\times6=12\) ft and the height of the triangle (height of the cone) is \(h = 15\) ft.
So, \(A=\frac{1}{2}\times12\times15\)
First, calculate \(12\times15 = 180\), then \(\frac{1}{2}\times180=90\) square feet.

Answer:

triangle; area \( = 90\space\text{ft}^2\)