Question

A cone is shown. The radius of the base circle is 6 feet. A line is drawn from the center point of the base circle to the apex and has a length of 15 feet. 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?

1) Triangle; area = 45 ft2
2) Triangle; area = 90 ft2
3) Circle; area = 36π ft2
4) Circle; area = 144π ft2

Answer 1

The cross section of the cone is a triangle, and its area is 90 ft², calculated using the formula for the area of a triangle (0.5 × base × height).

Step-by-step explanation:

The type of two-dimensional solid created by a vertical cross section of the cone that passes through the apex is a triangle. This is because the cross-section taken through the apex of a cone will always result in a triangle, one of the conic sections. To find the area of this triangular cross section, we can use the formula for the area of a triangle, which is A = 0.5 × base × height. In this case, the base of the triangle is the diameter of the cone's base circle, so it is 2 × radius (2 × 6 feet = 12 feet), and the height is the given line from the center to the apex, which is 15 feet.

Using the area formula, we get A = 0.5 × 12 ft × 15 ft, which simplifies to A = 90 ft². The correct answer to the question is therefore option 2: a triangle with an area of 90 ft².