Chapter 12.4, Problem 25PPS

a.

To determine

Complete the table.

Answer to Problem 25PPS

  $\begin{array}{ccccc}Color& \mathrm{Pr}obability& SectorArea& TotalArea& SectorArea/Totalarea\\ red& 0.2& 3.925& 19.625& 0.2\\ orange& 0.2& 3.925& 19.625& 0.2\\ yellow& 0.2& 3.925& 19.625& 0.2\\ green& 0.2& 3.925& 19.625& 0.2\\ blue& 0.2& 3.925& 19.625& 0.2\\ Total& 1& & & 1\end{array}$

Explanation of Solution

Given information:

A spinner has radius $2.5inches$

  

  

Calculation:

A spinner has radius $2.5inches$ , calculate area of circle,

  $\begin{array}{l}A=\pi r^2\\ =\pi \times {2.5}^2\\ =19.625\end{array}$

There are $5$ equal parts so probability of each color is,

  $P=\frac{A/5}{A}=0.2$

  $\begin{array}{ccccc}Color& \mathrm{Pr}obability& SectorArea& TotalArea& SectorArea/Totalarea\\ red& 0.2& 3.925& 19.625& 0.2\\ orange& 0.2& 3.925& 19.625& 0.2\\ yellow& 0.2& 3.925& 19.625& 0.2\\ green& 0.2& 3.925& 19.625& 0.2\\ blue& 0.2& 3.925& 19.625& 0.2\\ Total& 1& & & 1\end{array}$

b.

To determine

Make a conjecture about the relationship between the ratios of the area of the sector to the total area and probability.

Answer to Problem 25PPS

The ratio of the sector to the total area and probability of the spinner landing on each other is equal.

Explanation of Solution

Given information:

A spinner has radius $2.5inches$

  

Calculation:

As per the table, the ratio of the sector to the total area and probability of the spinner landing on each other is equal.

c.

To determine

Find the probability of a dart landing in each area of the board.

Answer to Problem 25PPS

The area of the circle is not equal so probability is not equally likely land at any point on the board.

Explanation of Solution

Given information:

A spinner has radius $2.5inches$

  

Consider and assume that any dart thrown will land on the board is equally likely to land at any point on the board.

  

Calculation:

  $\begin{array}{l}theradiusofdartboard=12-3=9\\ A=\pi r^2\\ =\pi 9^2\\ =81\pi \\ radiusinnercircle,r=3\\ A_i=\pi r^2\\ =\pi 3^2\\ =9\pi \\ radiusmiddlecircle,r=9-3=6\\ A_m=\pi 6^2-\pi 3^2\\ =27\pi \\ theareaofoutercircle,\\ A_o=\pi 9^2-\pi 6^2\\ =45\pi \end{array}$

Hence, the area of the circle is not equal so probability is not equally likely land at any point on the board.