A searchlight has a parabolic reflector that forms a "bowl," which is 18 inches wide from rim to rim and 10 inches deep. If the filament of the lightbulb is located at the focus, how far from the vertex of the reflector is it? Round your answer to the nearest hundredth. You may want to draw a graph to help you with this problem. Include it as part of your answer.
We can start by using the standard equation for a parabola with vertex at the origin:
y = 1/(4p) x^2
where p is the distance from the vertex to the focus. We want to find p, so we need to first find the equation of the parabola in terms of x and y.
Since the bowl is 18 inches wide, we know that the distance between the two points on the parabola that are 9 inches from the vertex in the x-direction is also 18 inches. This means that the x-coordinate of each of those points is ±9. Plugging this into the equation of the parabola gives us:
y = 1/(4p) (±9)^2 = 81/(4p)
Simplifying this gives us:
4py = 81
Now we need to use the fact that the bowl is 10 inches deep. This means that the y-coordinate of the point on the parabola that is 5 inches above the vertex is 10. Plugging this into the equation of the parabola gives us:
10 = 81/(4p) + c
where c is the y-coordinate of the vertex. But we know that the vertex is at (0, -10), so c = -10. Plugging this in and simplifying gives us:
40p = 71
So p = 71/40, which is approximately 1.78 inches. Therefore, the filament of the lightbulb is located 1.78 inches from the vertex of the reflector.
Here is a graph to help visualize the situation:
To solve this problem, we can use the equation for a parabola in vertex form:
y = a(x - h)^2 + k
Where (h, k) is the vertex of the parabola.
First, we need to find the equation of the parabola that represents the reflector. The width of the bowl is given as 18 inches, which means the distance between the two points on the parabola where y = 0 is 18 inches. We also know that the depth of the bowl is 10 inches. Since the filament is located at the focus which is at the vertex, k = 0.
Let's denote the vertex of the parabola as (h, 0). The equation for the parabola becomes:
y = a(x - h)^2
Now, to find the coefficient 'a' and the vertex (h, 0), we can use the information given in the problem:
1. The width of the bowl:
The distance between the two points where y = 0 is 18 inches. Since the parabola is symmetric, we can consider only one side. So, the distance from the vertex (h, 0) to a point on one side is 18/2 = 9 inches.
Using the point (x, 0) = (9, 0) and the vertex (h, 0), we can substitute these values into the equation of the parabola:
0 = a(9 - h)^2
2. The depth of the bowl:
The depth of the bowl is given as 10 inches. This means that the distance from the vertex (h, 0) to the bottom of the bowl is 10 inches.
Using the point (x, y) = (h, -10), we can substitute these values into the equation of the parabola:
-10 = a(h - h)^2
-10 = 0
Since the bottom of the bowl is at y = -10, we can conclude that the vertex of the parabola is also at y = -10.
Now, we have two equations:
0 = a(9 - h)^2
-10 = a(h - h)^2
-10 = 0
Since -10 is not equal to 0, we conclude that there is no value of 'a' that satisfies both equations. This means there is an error in the problem statement, as it is not possible for a parabolic reflector to have a width and depth that are described.
Without a valid problem statement, we cannot accurately determine the distance from the vertex of the reflector to the filament.
To find the distance from the vertex of the reflector to the filament, we will use the properties of a parabola.
1. Draw a graph: Start by drawing a coordinate system, where the x-axis represents the width of the parabolic reflector and the y-axis represents the depth. The vertex of the parabola is located at the origin (0, 0). Sketch a parabolic curve that opens upwards and passes through the points (9, 10) and (-9, 10). This represents the shape of the reflector.
2. Find the coordinates of the focus: The focus of the parabola is located along the axis of symmetry, which is the x-axis. The distance from the vertex to the focus is equal to the depth of the parabolic reflector, which in this case is 10 inches. Therefore, the coordinates of the focus are (0, 10).
3. Use the distance formula: The distance from the vertex of the reflector to the filament is the distance from the origin (vertex) to the focus. We can calculate this distance using the distance formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Plugging in the values, we have:
Distance = sqrt((0 - 0)^2 + (10 - 0)^2)
= sqrt(0 + 100)
= sqrt(100)
= 10 inches
So, the distance from the vertex of the reflector to the filament is 10 inches.