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?
To determine the distance from the bottom of the dish, we need to find the focal length of the parabolic dish. The focal length is the distance from the center of the dish to its focus point.
The formula for the focal length of a parabolic dish is given by:
f = (D^2) / (16d),
where f is the focal length, D is the diameter of the dish, and d is the depth of the dish.
Given:
D = 24 inches
d = 5 inches
Substituting the given values into the formula:
f = (24^2) / (16 * 5)
= 576 / 80
= 7.2 inches
Therefore, the microphone should be placed 7.2 inches from the bottom of the dish.
To determine the distance from the bottom of the dish where the microphone should be placed, we need to consider the properties of a parabolic reflector.
The equation of a parabola in standard form is y = ax^2, where (x, y) are the coordinates on the parabolic shape, and a is a constant.
In this case, the width of the reflective dish is given as 24 inches, which means the diameter of the parabolic shape would be 24 inches. Since the width is the distance across the opening, the maximum width of the dish is double the diameter, which is 2 * 24 = 48 inches.
To determine the value of a in the parabolic equation, we need the depth of the dish. The depth is given as 5 inches, which is the distance from the dish's opening to its deepest point.
Given that the depth of the dish is the distance from the vertex (the deepest point) to the focus (where the microphone should be placed), we can calculate the value of a.
The standard formula for the depth of a parabolic dish in terms of its width is: Depth = (1/16) * Width^2.
Substituting the values, we have: 5 = (1/16) * 48^2.
Simplifying the equation, we get: 5 = (1/16) * 2304.
To solve for (1/16), we multiply both sides of the equation by 16: (1/16) * 5 = (1/16) * (1/16) * 2304.
Simplifying further, we have: 5/16 = 2304/256.
Reducing the fractions to their simplest forms, we get: 5/16 = 9.
Therefore, the constant a in the parabolic equation is 9.
Now, we can determine the position of the microphone, which is the distance from the bottom of the dish to the focus. Using the equation y = ax^2, where y is the depth and x is the horizontal distance from the opening:
5 = 9 * x^2.
To find x, we need to rearrange the equation:
x^2 = 5/9.
Taking the square root of both sides, we get:
x = sqrt(5/9).
Simplifying the square root, we have:
x = sqrt(5) / sqrt(9).
The square root of 9 is 3, so we have:
x = sqrt(5) / 3.
Hence, the microphone should be placed at a distance of sqrt(5) / 3 inches from the bottom of the dish.
Place the vertex at (0,0)
Recall that the parabola x^2 = 4py has its focus at (0,p)
Now, since y = ax^2, and y(12) = 5, a=5/144
so now we have y = 5/144 x^2
That is, x^2 = 144/5 y
That makes p = 144/20 = 36/5
The focus 36/5 inches from the vertex.