Explanation:
The shape of a parabola is determined by its focus and directrix. In this problem, the focus is 30 ft above the ground, so the directrix is 30 ft below the ground. Since the ground is the x-axis, the directrix is a horizontal line with equation y = -30.
The base of the satellite dish should be positioned on the axis of symmetry of the parabola, which passes through the focus and is perpendicular to the directrix. Since the directrix is a horizontal line, the axis of symmetry is a vertical line passing through the focus. Therefore, the base of the satellite dish should be positioned directly below the focus, which is (0, 30).
The equation of a parabola with vertex at the origin, opening upwards or downwards, and axis of symmetry along the y-axis is given by the equation y = a x^2, where a is a constant. In this problem, the parabola opens upwards and its vertex is (0, 30), so the equation of the parabola is of the form y = a x^2 + 30.
To determine the value of a, we need to use the fact that the directrix is y = -30. The distance from a point (x, y) on the parabola to the directrix is given by |y - (-30)| = |y + 30|. The distance from the same point to the focus (0, 30) is given by the distance formula:
sqrt(x^2 + (y - 30)^2)
Since the point lies on the parabola, we can substitute y = ax^2 + 30 into this expression and simplify:
sqrt(x^2 + (a x^2)^2) = sqrt(1 + a^2) x^2
Now we can set the two expressions for the distances equal to each other and solve for a:
|ax^2 + 30 + 30| = sqrt(1 + a^2) x^2
|ax^2 + 60| = sqrt(1 + a^2) x^2
a^2 x^4 + 120 a x^2 + 3600 = x^4 + a^2 x^4
(a^2 - 1) x^4 + 120 a x^2 + 3600 = 0
This is a quartic equation in x^2. We can solve for x^2 using the quadratic formula:
x^2 = (-120 a ± sqrt(120^2 a^2 - 4 (a^2 - 1) 3600)) / 2(a^2 - 1)
x^2 = (-60 a ± sqrt(3600 a^2 + 14400)) / (a^2 - 1)
For the base of the satellite dish to be positioned on the ground, we need x^2 to be non-negative, which means the discriminant of the quadratic formula must be non-negative:
3600 a^2 + 14400 ≥ 0
a^2 ≥ -4
Since a is a positive constant that determines the shape of the parabola, we can take the positive square root of both sides and obtain a ≥ 0. Therefore, the equation of the satellite is:
y = a x^2 + 30, where a is a non-negative constant.
None of the provided equations match the answer obtained.