Question

Item 7 A parabolic spotlight is being designed with the lightbulb at the focus point. If the lightbulb is placed 2.5 inches above the vertex and the height of the spotlight is 10 inches, what is the width of the spotlight? Item 7 A parabolic spotlight is being designed with the lightbulb at the focus point. If the lightbulb is placed 2.5 inches above the vertex and the height of the spotlight is 10 inches, what is the width of the spotlight?

Answer 1

The width of the spotlight is 20 inches

Explanation:

The location of the bulb = 2.5 inches above the vertex

The height of the spotlight = 10 inches

The width of the spotlight =

The

The height of

The equation of a parabola is given as follows;

(x - h)² = 4p(y - k)

Focus, (h, k + p) = (0, 2.5)

y = k - p

k + p = 2.5

Vertex = (h, k) = (0, 0), therefore, k = 0

p = 2.5

We have;

(x - 0)² = 4p(y - 0)

x² = 4×2.5 × y = 10·y

Therefore, given that y = 10, we have;

x² = 10 × 10 = 100

x = √100 = ±10

The coordinates of the two end points are;

(-10, 10), and (10, 10)

Therefore, the width of the spotlight = the distance between the end points

Given that the y-values are the same, we have

Distance = 10 - (-10) = 10 + 10 = 20 inches

∴ The width of the spotlight = 20 inches.