Question

Area as probabilityA dart hits the square dartboard shown below at a random point. Find the probability that the dart lands in the shaded circular region. Each side of thedartboard is 14 in, and the radius of the shaded region is 6 in.Use the value 3.14 for 1. Round your answer to the nearest hundredth.

Answer 1

To answer this question, we will compute the compute both areas, and then we will use the following expression to determine the probability:

`\frac{\text{favorable cases}}{total\text{ cases}}.`

In this case, the favorable cases will be represented by the area of the circle, and the total cases will be represented by the area of the square.

Now, the area of a circle and a square can be determined with the following formulas:

`\begin{gathered} A_c=\pi r^2, \\ A_s=s^2, \end{gathered}`

where r is the radius of the circle and s is the side of the square.

Substituting r=6 in, and s=14 in in the above formulas, we get:

`\begin{gathered} A_c=\pi(6in)^2\approx36\pi in^2, \\ A_s=(14in)^2=196in^2. \end{gathered}`

Finally, using the expression for the probability, we get:

`P=(36\pi in^2)/(196in^2)\approx0.58.`

Answer:

`0.58.`