Question

a calculator is allowed for this question. a satellite dish is the shape of a paraboloid. the dish is 42 inches wide, and 10 inches deep. how many inches should the receiver be located from the vertex for optimal reception? (round to the nearest thousandth)

Explanation:

Step1: Set up the parabola equation

Place the vertex of the parabola at the origin $(0,0)$ and the parabola opening upwards. The standard - form equation of a parabola opening upwards is $x^{2}=4py$. The dish is 42 inches wide and 10 inches deep. When $y = 10$, $x=\pm21$ (since the width is 42 inches, half of it is 21 inches on each side of the y - axis).

Step2: Substitute the values of $x$ and $y$ into the equation

Substitute $x = 21$ and $y = 10$ into the equation $x^{2}=4py$. We get $(21)^{2}=4p\times10$.

Step3: Solve for $p$

First, calculate $(21)^{2}=441$. Then the equation becomes $441 = 40p$. Solving for $p$, we have $p=\frac{441}{40}=11.025$.

Answer:

$11.025$