An engineer designs a satellite dish with a parabolic cross section. The dish is 14 ft wide at the opening, and the focus is placed 4 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.
a) To find the equation of the parabola, based on the given information, we need to use the standard form equation for a parabola with its vertex at the origin: y^2 = 4px.
First, let's determine the value of p, which represents the distance from the vertex to the focus. In this case, p is given as 4 ft.
Since the dish is 14 ft wide at the opening, we know that the x-coordinate of the point where the parabolic curve intersects the x-axis is x = 7 ft.
Using this information, we can substitute the values into the equation y^2 = 4px to find the equation of the parabola:
(7)^2 = 4 * 4 * y
49 = 16y
y = 49/16
Therefore, the equation of the parabola is y^2 = (49/16)x.
b) To find the depth of the satellite dish at the vertex, we need to determine the y-coordinate of the vertex. Since the vertex is at the origin, the y-coordinate is 0.
So, the depth of the satellite dish at the vertex is 0 ft.
a) To find an equation for the parabola, we can use the vertex form of a parabola equation, which is given by:
y = a(x - h)^2 + k
Where (h, k) is the vertex of the parabola.
In this case, the vertex is given as (0, 4) since we are positioning the coordinate system with the origin at the vertex. The x-axis is the axis of symmetry, so the parabola is symmetric about the y-axis.
The width of the opening of the satellite dish is 14 ft, which corresponds to the distance between two points on the parabola with y = 0. Since these points are symmetric about the y-axis, we can say that x = 7 ft.
Substituting the vertex (0, 4) and the point (7, 0) into the equation, we can solve for 'a'.
0 = a(7 - 0)^2 + 4
0 = 49a + 4
49a = -4
a = -4/49
Thus, the equation of the parabola is:
y = (-4/49)x^2 + 4
b) The depth of the satellite dish at the vertex refers to the value of 'k' in the vertex form equation. From the equation above, we can see that k = 4. Therefore, the depth of the satellite dish at the vertex is 4 ft.
the general equation is
... x = a(y - k)^2 + h
the vertex is at the origin, so h and k are both zero
a = 1 / 4p
... where p is the distance from the vertex to the focus
b) the center depth is the value of x at the edges of the dish (where y equals ±7)