Questions: Let V be the volume of a right circular cone of height h=4 whose base is a circle of radius R=2. (a) Use similar triangles to find the area of a horizontal cross section at a height y. Give your answer in terms of y. (Use symbolic notation and fractions where needed.) A(y)=

Solution Steps

We need to find the area of a horizontal cross-section of a right circular cone at a height \( y \). The cone has a height \( h = 4 \) and a base radius \( R = 2 \).

The cross-section at height \( y \) forms a smaller, similar cone within the larger cone. The height of the smaller cone is \( y \), and its radius \( r(y) \) is proportional to \( y \).

Since the triangles are similar: \[ \frac{r(y)}{y} = \frac{R}{h} = \frac{2}{4} = \frac{1}{2} \] Thus, \[ r(y) = \frac{y}{2} \]

The area \( A(y) \) of the horizontal cross-section at height \( y \) is the area of a circle with radius \( r(y) \): \[ A(y) = \pi \left( r(y) \right)^2 = \pi \left( \frac{y}{2} \right)^2 = \pi \frac{y^2}{4} \]

Final Answer