Chapter 11.8, Problem 6WE

(a)

To determine

To Find: The percentage of darts that will hitting the board will land inside the inner square.

Answer to Problem 6WE

  $50\%$

Explanation of Solution

Given:

  

Let the side of outer square be $4x$

Therefore

The side of Inner Square is $2\sqrt{2}x$

The probability percentage of darts that will hitting the board will land inside the inner square.

  $\begin{array}{l}P=\frac{areaofinner}{areaofouter}\\ P=\frac{s_i{}^2}{s_o{}^2}\\ P=\frac{{\left(2\sqrt{2}x\right)}^2}{{(4x)}^2}\\ P=\frac{8x^2}{16x^2}\\ P=50\%\end{array}$

(b)

To determine

To Find: The percentage of darts that will hitting the board will land outside the inner square but inside the circle.

Answer to Problem 6WE

  $28.5\%$

Explanation of Solution

Given:

  

Area of circle

  $\begin{array}{l}$ A_c=\pi r^2=\pi {(2x)}^2$A_c=12.57x^2\end{array}$

Probability percentage of darts that will hitting the board will land outside the inner square but inside the circle.

  $\begin{array}{l}P=areaofcircle-\frac{areaofinner}{areaofouter}\\ P=12.57x^2-\frac{8x^2}{16x^2}\\ P=28.5\%\end{array}$