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# 6.2: A Modeling Example: Counting Ping Pong Balls

Created by: CK-12

Suppose you have a cylinder of height $h$ with base diameter $b$ (perhaps an empty pretzel jar), and you wish to know how many ping-pong balls of diameter $d$ have been placed inside the cylinder. How could you determine this? This problem, along with the strategy for computing the lower bound on the number of ping-pong balls, is adapted from Starfield (1994).

A lower bound for this problem is found as follows. Define the following variables:

• $N_L-$Lower bound on the number of balls that fit into the cylinder.
• $V_{cyl}-$The volume of the cylinder.
• $V_{cube}-$The volume of a cube that encloses a single ball.

$V_{cyl} & = h\pi(\tfrac{b}{2})^2 \\V_{cube} & = d^3$

The lower bound $N_L$ is found by dividing the volume of the cylinder by the volume of the cube enclosing a single ball:

$N_L = \tfrac{V_{cyl}}{V_{cube}}$

Exercise 11

You are given the following values:

• $d = 1.54\;\mathrm{in}$
• $b = 8\;\mathrm{in}$
• $h = 14 \;\mathrm{in}$

Type commands at the command line prompt to compute $N_L$.

Exercise 12

Create an m-file to solve Exercise 11.

To complicate your problem, suppose that you have not been given values for $d$, $b$, and $h$. Instead you are required to estimate the number of ping pong balls for many different possible combinations of these variables (perhaps $50$ or more combinations). How can you automate this computation?

One way to automate the computation of $N_L$ for many different combinations of parameter values is to use a for loop. The following exercises ask you to develop several different ways that for loops can be used to automate these computations.

Exercise 13

Add a for loop to your m-file from Exercise 12 to compute $N_L$ for $b = 8 \;\mathrm{in}$, $h = 14 \;\mathrm{in}$, and values of $d$ ranging from $1.0 \;\mathrm{in}$ to $2.0 \;\mathrm{in}$.

Exercise 14

Modify your m-file from Exercise 13 to plot $N_L$ as a function of $d$ for $b = 8 \;\mathrm{in}$ and $h = 14 \;\mathrm{in}$.

Exercise 15

Modify your m-file from Exercise 13 to compute $N_L$ for $d = 1.54 \;\mathrm{in}$ and various values of $b$ and $h$.

## Date Created:

Feb 23, 2012

Apr 29, 2014
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