Question

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. if the mirror is 34 feet across at its opening and is 12 feet deep, where will the heat source be concentrated?

the heat source will be concentrated about \\(\square\\) feet above the bottom of the mirror.
(round to two decimal places as needed.)

Explanation:

Step1: Define parabola equation

Let the parabola open upward with vertex at (0,0), so equation is $x^2 = 4py$, where $p$ is focus distance from vertex.

Step2: Identify point on parabola

Opening width is 34 ft, so half-width is 17 ft; depth is 12 ft. Point (17,12) lies on parabola.

Step3: Solve for p

Substitute (17,12) into equation: $17^2 = 4p(12)$ → $289 = 48p$ → $p = \frac{289}{48} ≈ 6.02$

Answer:

6.02