A satellite dish has a parabolic cross section and is 6 ft deep. The focus is 4 ft from the vertex. Find the width of the satellite dish at the opening. Round your answer to the nearest foot.
Thank you
To find the width of the satellite dish at the opening, we first need to understand the properties of a parabolic cross section.
1. The focus of a parabola is a fixed point inside the curve (in this case, the satellite dish).
2. The vertex is the lowest point on the curve, which is also where the axis of symmetry intersects the curve.
In this problem, the focus is given as 4 ft from the vertex and the depth of the dish is given as 6 ft.
Since the dish is symmetric, we can assume that the distance from the focus to the vertex is the same as the distance from the vertex to the opening (width) of the dish.
So, the total distance from the vertex to the width of the dish is 4 ft + x ft (where x is the width of the dish at the opening).
The total depth of the dish is given as 6 ft. Using this information, we can set up the equation:
4 ft + x ft + x ft = 6 ft
Combining like terms, we get:
2x + 4 ft = 6 ft
Subtracting 4 ft from both sides of the equation, we have:
2x = 2 ft
Finally, dividing both sides by 2, we find:
x = 1 ft
Therefore, the width of the satellite dish at the opening is approximately 1 foot.
To find the width of the satellite dish at the opening, we can use the equation for a parabolic cross-section, which is given by the equation:
y^2 = 4px
where y is the distance from the vertex to a point on the curve, p is the distance from the vertex to the focus, and x is the horizontal distance from the vertex to the point on the curve.
In this problem, we are given that the dish is 6 ft deep and the focus is 4 ft from the vertex. This means that p = 4.
To find the width at the opening, we need to find the x-coordinate of the point on the curve where y = 6.
Plugging in the values into the equation, we have:
6^2 = 4 * 4 * x
36 = 16x
x = 36/16
x = 2.25
Therefore, the width of the satellite dish at the opening is approximately 2.25 ft. Rounded to the nearest foot, the width is 2 ft.
from y^2 = 4px
the parabola has equation:
y^2 = 16x
at its widest point, x = 6
y^2 = 16(36) = 576
y + ±24
so the width = 48 ft