Chapter 14, Problem 14.1P

To determine

Probability that a skewer, jabbed at random into the pancake will hit a berry.

Answer to Problem 14.1P

Probability of a hit is $n_{tar}\sigma =0.027$

Explanation of Solution

Write the equation for area of cross section of blue berry.

  $\sigma =\pi {\left(\frac{d}{2}\right)}^2$

Here $d$ is the diameter of the blue berry.

Therefore,

  $\begin{array}{c}\sigma =\pi {\left(\frac{1\text{cm}}{2}\right)}^2\\ =0.79{\text{cm}}^2\end{array}$

Target density is given by

    $n_{\text{tar}}=\frac{\text{number}}{\text{area}}$

Area of target is,

  $\begin{array}{c}\pi r^2=\pi {\left(\frac{15\text{cm}}{2}\right)}^2\\ =176.625{\text{cm}}^2\end{array}$

Number of blue berries on the target is $6$

Thus,

    $\begin{array}{c}{n}_{\text{tar}}=\frac{6}{176.625{\text{cm}}^2}\\ =0.034{\text{cm}}^{\text{-2}}\end{array}$

Probability of a hit is,

  $\begin{array}{c}{n}_{tar}\sigma =\left(0.034{\text{cm}}^{-2}\right)\left(0.79{\text{cm}}^{\text{2}}\right)\\ =0.027\end{array}$

Conclusion:

Probability of a hit is $n_{tar}\sigma =0.027$