Chapter 8.6, Problem 69E

(a)

To determine

To find: the probability that the coin covers a vertex of one of the squares on the grid.

Answer to Problem 69E

The probability of the coin is $\frac{\pi }{4}$ for covering the vertex.

Explanation of Solution

Given information:

A coin of diameter $d$ is randomly dropped onto a paper that contains a grid of squares $d$ units on a side.

Given figure

  

Calculation:

Consider a point $O$ on the paper over which the center of coin will lie.

It is clear that the coin covers the vertex when the distance between $O$ and vertex is less than $d$

Since every gird on the paper forms four quarter circle

The radius of each circle is $\frac{d}{2}$

Therefore, the area of each circle is given by $\pi \frac{d^2}{4}$

Also, since the side of square is $d$ units

Therefore, the area of each grid is given by $d^2$

It is clear that $\frac{\pi }{4}$ of the area of grid is covered by the point $O$ which is less than $d$ units away from the vertex.

So, the probability of the coin is $\frac{\pi }{4}$ for covering the vertex.

(a)

To determine

To find: the probability that the coin covers a vertex of one of the squares on the grid.

Answer to Problem 69E

The probability of the coin is $\frac{\pi }{4}$ for covering the vertex.

Explanation of Solution

Given information:

A coin of diameter $d$ is randomly dropped onto a paper that contains a grid of squares $d$ units on a side.

Given figure

  

Calculation:

Consider a point $O$ on the paper over which the center of coin will lie.

It is clear that the coin covers the vertex when the distance between $O$ and vertex is less than $d$

Since every gird on the paper forms four quarter circle

The radius of each circle is $\frac{d}{2}$

Therefore, the area of each circle is given by $\pi \frac{d^2}{4}$

Also, since the side of square is $d$ units

Therefore, the area of each grid is given by $d^2$

It is clear that $\frac{\pi }{4}$ of the area of grid is covered by the point $O$ which is less than $d$ units away from the vertex.

So, the probability of the coin is $\frac{\pi }{4}$ for covering the vertex.

(b)

To determine

To approximate: the value of $\pi$ by performing the experiment 100 times.

Answer to Problem 69E

The area covered by the coin in 100 drops is $25\pi$.

Explanation of Solution

Given information:

A coin of diameter $d$ is randomly dropped onto a paper that contains a grid of squares $d$ units on a side.

Given figure

  

Calculation:

In the previous part as it is shown that dropping a coin covers $\frac{\pi }{4}$ area of the grid.

Since the coin is dropped 100 times.

So the area covered by the coin in 100 drops is

  $100\times \frac{\pi }{4}=25\pi$