Question

The mirror in the figure shown is in the shape of a parabolic surface. it will be used to concentrate the rays of the sun at its focus, creating a heat source. The diameter of the mirror is 24 inches and it depth is 4 inches. How far from the base of the mirror shoud the heat source be concentrated? Show all work.

Answer 1

9 inches

Step-by-step explanation

We have to find the height at which the heat source should be. If we look at the mirror sideways we have a parabola,

For the rays to be concentrated at the same point, the heat source must be placed at the focus of the parabola.

As we can see, if we place the vertex of the parabola at the point (0, 0), the equation is,

`\begin{gathered} (x-h)^2=4p(y-k) \\ \\ x^2=4py \end{gathered}`

The y-coordinate of the focus is the value of p. To find it, we know that the parabola passes through the points (-12, 4) and (12, 4). Using any of these points in the equation above, we can find p,

`\begin{gathered} 12^2=4p\cdot4 \\ 144=16p \\ p=(144)/(16)=9 \end{gathered}`

Hence, the heat source should be concentrated at 9 inches from the base of the mirror.