A parabolic microphone used on the sidelines of a professional football game uses a reflective dish 24 inches wide and 5 inches deep. How far from the bottom of the disk should the microphone be placed?
a.) 6.8 in
b.) 7 in
c.) 7.2 in
d.) 7.4 in
To solve this problem, we can use the properties of a parabola.
First, let's draw a diagram:
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Here, the dish represents a parabolic shape with a width of 24 inches and a depth of 5 inches. The microphone needs to be placed at a certain distance from the bottom of the dish. Let's call this distance "x".
The equation of the parabola can be written in vertex form as: y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola and a represents the width of the dish.
Since the dish has a width of 24 inches, we can set a = 4 (since 2 * 4 = 8, and the parabolic shape extends 8 units on either side of the vertex point).
The vertex of the parabola is the lowest point on the dish, located at the center. In this case, the vertex would be (0, -5) since the dish is 5 inches deep.
Plugging these values into the vertex form equation, we have:
y = 4(x - 0)^2 - 5
Now, let's substitute y with x since we need to find the distance from the bottom of the dish:
x = 4(x - 0)^2 - 5
Simplifying the equation:
x = 4x^2 - 5
Rearranging the equation:
4x^2 - x - 5 = 0
We can solve this quadratic equation by factoring or using the quadratic formula. Factoring is not possible in this case, so let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 4, b = -1, and c = -5. Plugging these values into the quadratic formula:
x = (-(-1) ± √((-1)^2 - 4 * 4 * -5)) / (2 * 4)
Simplifying:
x = (1 ± √(1 + 80)) / 8
x = (1 ± √81) / 8
Taking the square root:
x = (1 ± 9) / 8
This gives us two possible solutions: x = (1 + 9) / 8 = 10 / 8 = 1.25 inches and x = (1 - 9) / 8 = -8 / 8 = -1 inch.
Since the distance from the bottom of the dish cannot be negative, we can discard the negative solution.
Therefore, the microphone should be placed approximately 1.25 inches from the bottom of the dish.
However, none of the answer choices provided match this calculation. It's possible that there might be a mistake in the problem or in the available answer choices.