Chapter 1.7, Problem 35E

Solution

To calculate the height of the cardboard cylinder that need be to fit in the middle of the dish.

The height of the cylinder $=6\text{ in}$

Given:

Diameter of a parabolic dish $=32\text{ in}$

Depth of a parabolic dish $=8\text{ in}$

Radius of the cardboard cylinder $=8\text{ in}$

Calculation :

The general equation of the parabola is

  $y=k{x}^2$

Now, this parabola function should pass through the points $\left(16,8\right)$ and $\left(-16,8\right)$ , i.e. parabola should satisfy the points:

  $\begin{array}{l}y=8\\ x=\pm 16\end{array}$

Now,

  $\begin{array}{l}8=k{\left(\pm 16\right)}^2\\ k=\frac{8}{256}\\ k=\frac{1}{32}\end{array}$

So,

  $y=\frac{1}{32}{x}^2$

Now, to find the height of the carboard cylinder, $y-$ coordinate of the parabola from the centre need to be calculated at $x=8$ :

  $\begin{array}{l}y=\frac{1}{32}{\left(8\right)}^2\\ y=\frac{8\times 8}{32}\\ y=2\text{ in}\end{array}$

Hence, from that point to the top of the dish, the height of the cardboard cylinder is

Height $=8-2=6\text{ in}$

Graph: