The reflecting dish of a parabolic microphone has a cross-section in the shape of a parabola. The microphone itself is placed on the focus of the parabola. If the parabola is 60 inches wide and 30 inches deep, how far from the vertex should the microphone be placed?
If we set the vertex at (0,0) then the curve goes through (30,30)
y = 1/30 x^2
x^2 = 30y
Since the parabola x^2 = 4py has its focus at y=p, 4p=30 and the mic should be 15/2 inches up from the vertex.
To determine the distance from the vertex to the microphone placement, we need to find the focus of the parabolic shape.
A parabola has a specific equation: y = ax^2 + bx + c. In this case, the parabolic shape has a cross-section in the shape of a parabola opening upward, so we have a positive "a" value.
The standard equation for a parabola with its vertex at the origin is given by:
y = ax^2
Since the vertex of this parabola is at (0,0), we can substitute these values into the equation:
0 = a(0)^2
0 = a(0)
0 = 0
From the given dimensions, we know that the width at the opening of the parabola is 60 inches. Since the parabolic shape is symmetric, there will be a width of 30 inches on either side. Therefore, the parabola's width can be represented as the distance from the vertex to either end, which is 30 inches.
To determine the focus of the parabola, we can use the equation c = a/4f, where c is the distance from the vertex to the focus, and f is the focal length.
In our case, the width of the parabola is given as 60 inches, so we can substitute this value into the equation:
60 = a/4f
From this equation, we can solve for a:
a = 240f
Now, we can substitute a = 240f into the equation of the parabola y = ax^2 to find the value of c:
y = 240f(x)^2
Since the microphone is placed at the focus, the y-coordinate of the focus is equal to c:
c = 240f(0)^2
c = 0
This means that the focus of the parabolic shape is at y = 0. Therefore, the microphone should be placed at a distance of 0 inches from the vertex, which means the microphone should be placed right at the vertex.
To find the distance from the vertex to the microphone, we need to understand the properties of a parabola.
The standard form equation of a parabola with a vertical axis is given by:
y = a(x-h)^2 + k
Where (h, k) represents the vertex of the parabola. In this case, the dimensions of the parabolic dish are given as 60 inches wide and 30 inches deep.
Since the parabola is symmetric with respect to its vertex, we know that the width is 2a, where a is a positive constant. Therefore, 2a = 60 inches, which means a = 30 inches.
Next, we need to find the coordinates of the vertex (h, k). The vertex lies at the deepest point of the parabola, which is at the focus of the parabolic microphone. Since the microphone is placed at the focus, it will be at a distance of "k" from the vertex.
We know that the equation of a parabola in standard form has its vertex at (h, k), where h is the x-coordinate and k is the y-coordinate of the vertex. In this case, the vertex is (h, 0).
To find "h," we know that the vertex is also the midpoint of the focal distance. The focal distance is equal to a, which is the coefficient of the squared term in the equation.
In this case, the vertex (h, 0) is the midpoint of the focal distance, thus h = a/2 = 30/2 = 15 inches.
Finally, the microphone should be placed at a distance of "k" from the vertex, which is 0 in this case.
Therefore, the microphone should be placed 0 inches from the vertex.
To summarize:
- The equation of the parabola is y = 30(x - 15)^2
- The vertex of the parabola is (15, 0)
- The microphone should be placed at a distance of 0 inches from the vertex.