Chapter 8.2, Problem 58PS

Solution

To determine : Equation for the cross section of dish opening upward with its vertex at the origin and its depth.

The required equation is $x^2=36y$ and the depth is $4.34\text{ meters}$ .

Given information :

A Very Large Array in New Mexico consists of $27$ radio telescopes. For each parabolic telescope dish, the diameter is $25$ meters and the distance between the vertex and focus is $0.36$ times the diameter.

Formula used:

The standard form of equation of a parabola with vertex $\left(0,0\right)$ , focus $\left(0,p\right)$ is given by,

  $x^2=4py$

Explanation :

The distance $\left(p\right)$ between vertex and focus will be,

  $\begin{array}{c}p=25\times 0.36\\ =9\text{ meters}\end{array}$

The standard form of equation of a parabola opening upward with vertex $\left(0,0\right)$ is given by,

  $x^2=4py$

Substitute $p=9$ .

  $\begin{array}{c}{x}^2=4\left(9\right)y\\ x^2=36y\end{array}$

The diameter is $25$ meters, so the radius will be,

  $\begin{array}{c}\text{Radius}=\frac{25}{2}\frac{dy}{dx}\\ =12.5\text{ meters}\end{array}$

Thus, the dish extends $12.5\text{ meters}$ to both sides of the vertex $\left(0,0\right)$ .

Now, substitute $x=12.5$ in $x^2=36y$ .

  $\begin{array}{c}{\left(12.5\right)}^2=36y\\ 156.25=36y\\ y=\frac{156.25}{36}\\ =4.34\text{ meters}\end{array}$

Therefore, the required equation is $x^2=36y$ and the depth is $4.34\text{ meters}$ .