Chapter 10.2, Problem 71AYU

Solar Heat A mirror is shaped like a paraboloid of revolution and will be used to concentrate the rays of the sun at its focus, creating a heat source. See the figure. If the mirror is 20 feet across at its opening and is 6 feet deep, where will the heat source be concentrated?

To determine

To find: Where the heat will be concentrated if the mirror is 20 feet across at its opening and 6 feet deep.

Answer to Problem 71AYU

Heat will be concentrated about $4.17\text{feet}$ from the base.

Explanation of Solution

Given:

A mirror is shaped like a paraboloid of revolution and will be used to concentrate the rays of the sun at its focus, creating a heat source. The mirror is 20 feet across at its opening and is 6 feet deep.

Formula used:

Vertex	Focus	Directrix	Equation	Description
$\left(0,0\right)$	$\left(0,a\right)$	$y=-a$	$x²=4ay$	Parabola, axis of symmetry is $y\text{-axis}$, opens up

Calculation:

Let the vertex of the parabola be $\left(0,0\right)$ and it opens up.

The equation of the parabola is given by $x²=4ay$ where a is the distance from vertex to focus that is location of the source.

The parabola is 20 feet across and 6 feet deep.

Hence $\left(10,6\right)$ and $\left(-10,6\right)$ lies on the parabola.

To solve for $a$, we substitute $\left(10,6\right)$ in the equation $x²=4ay$.

$10²=4*a*6$

$a=\frac{100}{24}=4.1666667\approx 4.17 \text{feet}$

Hence, the heat will be concentrated about $4.17\text{feet}$ from the base, along the axis of symmetry.