Final answer:
To create a parabolic mirror that concentrates sunlight on a pipe 2 inches from the vertex, the equation of the parabola is x² = 8y. This equation reflects the parabolic shape where the mirror will focus sunlight onto the pipe located at the focal point.
Step-by-step explanation:
The student is looking to write the equation for a parabola that models the cross section of a mirror designed to focus sunlight on a pipe. Given that the pipe is located at the focus, 2 inches from the vertex of the mirror and the parabola opens upward, the standard form of a parabolic equation can be used, which is (x - h)² = 4p(y - k), where (h,k) is the vertex of the parabola, and p is the distance from the vertex to the focus. Working with inches, since the focus is 2 inches from the vertex and the parabola opens upward, p = 2. Therefore, the vertex is at the origin (0,0), leading to the simplified equation x² = 8y to model the parabola.
The concept of a parabolic mirror concentrating sunlight onto a focal point is applied in many practical scenarios, such as the generation of electricity using parabolic trough collectors in southern California. Understanding how the focal length (f) of a parabola relates to the radius (R) of a cylindrical section of a mirrored surface is pivotal in harnessing solar energy. In general, the radius of a mirrored surface designed to collect sunlight is twice the focal length (R = 2f), which is important for determining the amount of insolation focused onto a device.