Chapter 9, Problem 14PS

To determine

To find: the probability that hits the region outside the triangle.

Answer to Problem 14PS

The probability that the dart hits the region outside the triangle is 0.6815.

Explanation of Solution

Given information:

The dart is equally likely to hit any point inside the target.

Given figure

Calculation:

Note that the radius of the circle

  $r=6$

Therefore, the area of the circle target is given by

  $\begin{array}{c}A=\pi r^2\\ =3.14{\left(6\right)}^2\\ =3.14\times 36\\ =113.04\end{array}$

Also, the height of the triangle is 6 and the base of the triangle being the diameter of the circle is 13

Therefore, the area of the triangle is

  $\frac{1}{2}\left(12\right)\left(6\right)=36$

Therefore, the area of the circle outside the triangle is

  $113.04-36=77.04$

Therefore, the probability that the dart hits the region outside the triangle

  $\begin{array}{c}\frac{\text{area}\text{of}\text{the}\text{circle}\text{outside}\text{the}\text{triangle}}{\text{area}\text{of}\text{the}\text{circle}}\text{=}\frac{\text{77}\text{.04}}{\text{113}\text{.04}}\\ \text{=0}\text{.6815}\end{array}$

Hence, the probability that the dart hits the region outside the triangle is 0.6815.