Chapter 8.2, Problem 55PS

Solution

To calculate : The depth of the cross section of the trough.

The depth of the cross section of the trough is about $3$ feet.

Given information :

Solar energy can be concentrated using long troughs that have a parabolic cross section. The collected energy's uses include heating buildings, producing electricity, and producing fresh water from seawater.

  

Calculation :

Identify the focus of the parabola.

From the graph, it is clear that the focus lies $6$ units up from the origin. Thus, the focus of the given parabola is $\left(0,6\right)$ .

The standard form of the equation of a parabola with vertex at $\left(0,0\right)$ is $x^2=4py$ , with focus at $\left(0,p\right)$ . The value of $p$ is $6$ .

Substitute $6$ for $p$ in the standard form of the equation of a parabola.

  $\begin{array}{c}{x}^2=4\left(6\right)y\\ =24y\end{array}$

An equation for the cross section of the trough is $x^2=24y$ .

The depth is the $y-$ value at the antennas outside edge. The parabolic cross section extends $\frac{17}{2}\text{ or 8}\text{.5}$ feet to both sides of the vertex $\left(0,0\right)$ .

Substitute $8.5$ for $x$ in the equation $x^2=24y$ .

  $\begin{array}{l}{\left(8.5\right)}^2=24y\\ 72.25=24y\end{array}$

Divide both the sides by $24$ .

  $\begin{array}{l}\frac{72.25}{24}=\frac{24y}{24}\\ y\approx 3\end{array}$

Therefore, the depth of the cross section of the trough is about $3$ feet.