Chapter 8.2, Problem 9GP

Solution

To calculate : The depth of antenna using the equation for the cross section of the antenna with its vertex at origin and its focus $10$ feet to the right of its vertex.

The depth of antenna is about $18$ feet.

Given information :

A parabolic microwave antenna is $16$ feet in diameter.

Calculation :

Given depth of the parabolic microwave antenna is $16$ feet in diameter.

Write an equation for the cross section. Since the focus is to the right of the vertex $\left(0,0\right)$ , $p$ is positive. Thus, $p=10$ .

The standard form of the equation of a parabola with vertical axis of symmetry is $y^2=4px$ .

Substitute $10$ for $p$ in the equation and evaluate.

  $\begin{array}{c}{y}^2=4\left(10\right)x\\ =40x\end{array}$

The equation for the cross section is $y^2=40x$ .

Find the depth of the antenna.

The depth is the $y-$ value at the antennas outside edge. Since the antenna is $16$ feet in diameter, the antenna extends $\frac{16}{2}\text{ or 8}$ feet to either side of the vertex $\left(0,0\right)$ .

Substitute $8$ for $x$ in the equation $y^2=40x$ .

  $\begin{array}{c}{y}^2=40\left(8\right)\\ =320\end{array}$

Take the square root on both the sides.

  $\begin{array}{c}\sqrt{y^2}=\sqrt{320}\\ y\approx 18\end{array}$

Therefore, the depth of antenna is about $18$ feet.