Question

You can make a solar hot-dog cooker by shaping foil-lined cardboard into a parabolic trough and passing a wire through the focus of each end piece. How far from the bottom should the wire be placed?

Answer 1

The wire should be placed at a height of 2.25 inches from the bottom

Explanation:

With the assumption that the shape of the solar hot-dog cooker is a parabola. we have the equation for the shape given as y = a·x² + b·x + c

Whereby the coordinates of the vertex is taken as (h, k) = (0, 0), we have;

c = 0

∴ y = a·x² + b·x

From the diagram, we have;

When x = 6, y = 4

When x = -6, y = 4

Which gives;

4 = a·(-6)² + b·(-6) = 36·a - 6·b

4 = 36·a - 6·b.........................................(1)

4 = a·(6)² + b·(6) = 36·a + 6·b

4 = 36·a + 6·b........................................(2)

Subtracting equation (1) from equation (2) gives;

4 - 4 = 36·a - 6·b - (36·a + 6·b) = -12·b

0 = -12·b

b = 0

Therefore;

4 = 36·a - 6×0

a = 4/36 = 1/9

The focus = F(h, k + p)

p = 1/(4·a) = 1/(4 × 1/9) = 1/(4/9) = 9/4 in = 2.25 inches

The coordinates of the focus, F = (h, k + p) = (0, 9/4) = (0, 2.25)

Therefore, the wire should pass through the focus and placed at 2.25 inches above the bottom.