Chapter 11.8, Problem 15WE

(a)

To determine

To Show: The probability that the coin does not touch coin is ${\left(\frac{30-2R}{30}\right)}^2$

Explanation of Solution

Given: Suppose the coin with radius $R$

  

Probability the coin does not touch coin.

  $\begin{array}{l}P=\frac{\text{area of shaded region}}{\text{area of square}}\\ P=\frac{{(30-2R)}^2}{{(30)}^2}\\ P={\left(\frac{30-2R}{30}\right)}^2\end{array}$

Hence proved.

(b)

To determine

To Find: The value of $R$.

Answer to Problem 15WE

  $7.5mm$

Explanation of Solution

Given: Probability is $0.25$

  $\begin{array}{l}P={\left(\frac{30-2R}{30}\right)}^2\\ 0.25={\left(\frac{30-2R}{30}\right)}^2\\ \frac{30-2R}{30}=0.5\\ 30-2R=30(0.5)\\ 30-2R=15\\ 2R=15\\ R=7.5mm\end{array}$