An engineer designs a satellite dish with a parabolic cross section. The dish is 10 ft wide at the opening, and the focus is placed 8 ft from the vertex.
a) Position a coordinate system with the origin
at the vertex and the x-axis on the parabola's
axis of symmetry and find an equation of the
parabola.
b) Find the depth of the satellite dish at the vertex.
place the vertex at (0,0) and the focus at (8,0)
The directrix will be x = -8
let P(x,y) be any point
By definition:
√( (x-8)^2 + y^2) = x+8
square both sides
x^2 - 16x + 64 + y^2 = x^2 + 16x + 64
y^2 = 32x
so at the point where the dish is 10 ft wide, y = 5
25 = 32x
x = 25/32
a) Well, well, looks like we have a math question here. Let's dive right into it. Now, since we have the opening width and the focus of the parabolic dish, we can get cracking on that equation.
In our coordinate system, let's set the vertex as the origin. Now, since the x-axis is on the parabola's axis of symmetry, we can simplify our equation. The focus typically sits on the axis of symmetry of a parabola, so we know the focus will be located at (8,0).
We also know that the width of the dish at the opening is 10 ft. So, if we assume the vertex is (0,0) and the focus is at (8,0), we can take half of the width to get the distance from the vertex to the opening, which is 5 ft.
Now, let's think about the parabolic equation, which takes the form y = ax^2. Since our focus is located at (8,0), we know that every point (x,y) on the parabola should satisfy the distance formula from the vertex to the focus. In this case, it is sqrt((x - 8)^2 + y^2) = 5.
Now it's time to solve for y. We square both sides of our equation like a mad scientist, and we end up with (x - 8)^2 + y^2 = 25. And there you have it, my friend! The equation of the parabola is (x - 8)^2 + y^2 = 25. Let's move on to our next challenge!
b) Ah, the depth of the satellite dish at the vertex, you say? Well, if we refer back to our equation (x - 8)^2 + y^2 = 25, the vertex would be located at (8,0). So the depth of the dish at the vertex would simply be the y-coordinate of the vertex. And guess what, my friend? It's a big fat zero! The dish is flat at the vertex. So the depth of the satellite dish at the vertex is a resounding zero!
a) To find the equation of the parabola, we need to determine the vertex form of the equation.
First, let's place the coordinate system with the origin at the vertex (0,0) and the x-axis on the parabola's axis of symmetry.
The general vertex form equation for a parabola is given by y = a(x-h)^2 + k, where (h,k) is the vertex of the parabola and a is a constant determining the shape of the parabola.
Since the focus is placed 8 ft from the vertex, the x-coordinate of the focus is 8. This means that the vertex is halfway between the origin and the focus, which gives us the vertex coordinates as (4, 0).
Now we need to find the value of a.
The distance from the vertex to the opening of the dish is 10 ft. Since the opening is on the x-axis, this gives us the point (10/2, 0) = (5, 0) on the parabola.
Using the distance formula, we can find the distance from the focus to this point:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((8 - 5)^2 + (0 - 0)^2)
= sqrt(9)
= 3
Since a is the reciprocal of 4 times the distance from the focus to the vertex, we have:
a = 1 / (4 × 3)
= 1/12
Therefore, the equation of the parabola in vertex form is:
y = (1/12)(x - 4)^2
b) To find the depth of the satellite dish at the vertex, we need to determine the y-coordinate of the vertex.
Substituting the vertex coordinates (4, 0) into the equation of the parabola, we have:
0 = (1/12)(4 - 4)^2
0 = (1/12)(0)^2
0 = 0
Thus, the depth of the satellite dish at the vertex is 0 ft.
To answer these questions and find the equation of the parabola for the satellite dish, we need to understand the properties of a parabola and how it is defined mathematically.
A parabola is a U-shaped curve that is symmetric about its axis of symmetry. It can be defined either in vertex form or standard form. In this case, we can use the vertex form because we have the coordinates of the vertex.
Let's start with part (a) - finding the equation of the parabola.
1. Position a coordinate system with the origin at the vertex and the x-axis on the parabola's axis of symmetry:
- Place the vertex at the origin (0,0).
- Align the x-axis along the axis of symmetry.
2. The width of the satellite dish's opening is given as 10 ft. Since the parabola is symmetric, we know that the width from one side to the other is 20 ft.
3. The focus of the parabola is placed 8 ft from the vertex. The focus is a point on the parabola, and its distance from the vertex is known as the focal length.
Now, let's find the equation of the parabola in vertex form. The vertex form of the equation is given by:
y = a(x - h)^2 + k
where (h,k) represents the vertex of the parabola.
In this case, the vertex is at (0,0), so the equation becomes:
y = ax^2
To find the value of 'a', we need to use the focal length. In a parabola, the equation linking the focal length (f) to 'a' is:
f = 1/(4a)
Substituting the given values into the equation:
8 = 1/(4a)
Multiply both sides by 4a:
32a = 1
Divide both sides by 32:
a = 1/32
So, the equation of the parabola becomes:
y = (1/32)x^2
Now, let's move to part (b) - finding the depth of the satellite dish at the vertex.
The depth of the satellite dish at the vertex is the y-coordinate of the vertex. In this case, the vertex is at (0,0), so the depth is 0 ft. Therefore, the depth of the satellite dish at the vertex is 0 ft.
In summary:
a) The equation of the parabola is y = (1/32)x^2.
b) The depth of the satellite dish at the vertex is 0 ft.