I cannot figure out how to solve and graph this problem. You are desiginnig a parabolic dish to use for cooking on a camping trip. You plan to make the dish 60cm wide and 30cm deep. Where should you locate the cooking grill so that all of the light that enters the parabolic dish will be reflected toward the food? How does this point relate to the definiton of a parabola?
We can orient the dish so that the vertex is at the origin and the parabola opens upward...
If the dish is 60cm wide and 30cm deep, the points (-30, 30) and (30, 30) are on the graph
So......the form we have when the vertex is at the origin is this :
4py = x^2 where p is the distance between the origin and the focal point
And since (30,30) is on the graph, we have
4p (30) = 30^2
120p = 900 divide both sides by 120
p = 900/120
p = 15/2
So.....the cooking grill should be located at (0, 0 + p) = (0, 15/2)
This fits the definition of the parabola because any light waves striking the interior of the dish will be reflected towards the focal point.
The function is
4 (15/2)y = x^2 ⇒ 30y = x^2 ⇒ y = (1/30)x^2
Here is the graph with the focal point :