Problem #$3$

A cylindrical, vertical water tank has hemispheric end caps as shown. The radius of the cylinder and the caps is $r=40cm,$ and the length of the cylindrical part is $1\mathrm{.}2m.$

Hint:

For a hemispherical "concave up$"$ bowl of radius $R,$ the volume of water contained in it at a water level height $y$ is:

$V=\frac{}{3}{y}^2(3R-y)$

You are interested in developing a program to find the volume of water contained in the tank at a given height $h.$

Required Tasks

Write a MATLAB function named calculate$\_$volume that fulfills these specs:

Accepts the radius $r($in $m),$ cylinder height $($in $m),$ and the height $h($in $m)$ as input arguments.

Calculates the volume of water inside the tank if the water level is $h.$

Hint: Think of the tank as three distinct regions, each with its own set of volume equations.

While you could put all of the selection and calculation code within calculate$\_$volume, consider writing additional functions to make your program more modular or efficient.

Returns the computed volume as the output argument.

Create a MATLAB script that tests your function. The specifications are:

Within a loop:

Displays a menu of which allow the user to select from these options: enter a single $h$ value, or plot a graph of volume versus height $h,$ or quit the program.

If requested, calls your calculate$\_$volume function to do the actual calculation and then displays the result.

If the user wants a plot, then it should be generated like this:

Plots water volume versus height $h,$ where $h$ goes from $0m$ to the full height of the tank.

Has reasonable axes labels and a title.

Do NOT use element$-$by$-$element operators for this problem. Instead, use a for loop to create the vectors needed to make the plot.

Save your script file using this file name: hw$4$x$3.$m