In a factory, a parabolic mirror to be used in a searchlight was placed on the floor. It measured 40 centimeters tall and 90 centimeters wide. Find the equation of the parabola.
I tried working this problem out and I got y=-8/405x^2+40, but I'm not sure if it is right. Could someone please help me?
Is my answer correct then??
To find the equation of the parabola, we need to know the coordinates of its focus and directrix.
Given:
Height of the parabolic mirror = 40 centimeters
Width of the parabolic mirror = 90 centimeters
We know that the vertex of the parabola is at the origin (0, 0). Let's assume that the focus is located at (0, p).
To find the value of p, we can use the fact that the distance from the vertex to the focus is equal to the distance from the vertex to the directrix.
The distance from the vertex (0, 0) to the focus (0, p) is p.
The distance from the vertex (0, 0) to the directrix can be calculated using the equation:
Distance = (height of the parabolic mirror) / 2 = 40 / 2 = 20 centimeters.
Since the directrix is a vertical line, the equation of the directrix can be written as x = -20.
For a parabola, the distance from a point (x, y) to the directrix is given by the formula:
Distance = |y - (-20)| = |y + 20|
Setting the distances from the vertex to the focus and to the directrix equal, we get:
p = |y + 20|
To eliminate the absolute value, we can square both sides:
p^2 = (y + 20)^2
Now, let's consider a point (x, y) on the parabola.
The distance from the point (x, y) to the focus (0, p) can be calculated using the distance formula:
Distance = √((x - 0)^2 + (y - p)^2) = √(x^2 + (y - p)^2)
We know that the distance from the point (x, y) to the directrix x = -20 is equal to the distance from the point to the focus, so we have:
√(x^2 + (y - p)^2) = p + 20
Squaring both sides to eliminate the square root, we get:
x^2 + (y - p)^2 = (p + 20)^2
Expanding and simplifying the equation, we have:
x^2 + y^2 - 2py + p^2 = p^2 + 40p + 400
Simplifying further, we get:
x^2 + y^2 - 2py = 40p + 400
Dividing both sides of the equation by p, we have:
(x^2/p) + y^2/p - 2y = 40 + 400/p
As per the given dimensions of the mirror, the width is 90 centimeters. Since the width is equal to the focal length (2p), we have:
2p = 90
p = 90/2
p = 45
Substituting the value of p = 45 into the equation, we get:
(x^2/45) + y^2/45 - 2y = 40 + 400/45
Simplifying further, we have:
(x^2/45) + y^2/45 - 2y = 40 + 360/9
(x^2/45) + y^2/45 - 2y = 40 + 40
(x^2/45) + y^2/45 - 2y = 80
Multiplying both sides of the equation by 45 to eliminate fractions, we get the final equation of the parabola:
x^2 + y^2 - 90y = 3600
Therefore, the equation of the parabola is x^2 + y^2 - 90y = 3600.
To find the equation of a parabola, we need to know the vertex and one other point on the parabola. In this case, we can assume that the vertex is at the origin (0, 0) since the parabolic mirror is placed on the floor.
We are given the measurements of the mirror: height = 40 centimeters and width = 90 centimeters.
To find the equation of the parabola, we need to find the point at the edge of the mirror. Since the mirror is symmetric, we can find the point on one side and assume that the same point exists on the other side.
Let's consider the top-right corner of the mirror. The x-coordinate of this point will be half of the width, and the y-coordinate will be the height. So the coordinates of this point are (45, 40).
Now we have two points: the vertex (0, 0) and (45, 40). We can use these points to find the equation of the parabola.
Using the general form of a parabola equation: y = ax^2 + bx + c, we can substitute the coordinates of the two points to get two equations:
For the vertex (0, 0):
0 = a(0)^2 + b(0) + c
0 = c
For the point (45, 40):
40 = a(45)^2 + b(45) + c
Since we found that c = 0 in the first equation, we can rewrite the second equation as:
40 = 45^2a + 45b
Now we have two equations:
0 = c (equation 1)
40 = 45^2a + 45b (equation 2)
From equation 1, we know that c = 0, and substituting this value into equation 2 gives:
40 = 45^2a + 45b
Simplifying equation 2:
40 = 2025a + 45b
Divide both sides by 45:
40/45 = 2025a/45 + b
4/9 = 45a/45 + b
4/9 = a + b
So, the equation of the parabola is:
y = ax^2 + bx + c
y = ax^2 + bx + 0
y = ax^2 + bx
y = (4/9)x^2 + (4/9)x
Hence, the equation of the parabola is y = (4/9)x^2 + (4/9)x.
draw a picture of the parabola, centered over (0,0)
vertex: (0,40)
x-intercepts: x = 45, -45
y = (45^2 - x^2) * 40/45^2
you can pretty it up if you want.