Chapter 11.6, Problem 23PSC

To determine

To calculate: The probability that the point will be in shaded region of interior of circle with center P .

Answer to Problem 23PSC

The probability thatthe point will be in shaded region is $\frac{33}{49}$ .

Explanation of Solution

Given information:

Circles with center O and center P are tangent to x axis at point Q .

Formula used:

Area of a circle: $A=\pi r^2$

r = radius of circle.

  $P\left(E\right)=\frac{Numberoffavourableoutcomes}{TotalNumberofoutcomes}$

Calculation:

  

Circle with center O will have radius 4 because its y coordinate is 4.

Larger circle will have radius 7 because its y coordinate is 7.

Area of smaller circle $=\pi {(4)}^2=16\pi$

Area of larger circle $=\pi {(7)}^2=49\pi$

The area of the shaded region is $49\pi -16\pi$ or $33\pi$ .

Probability means possibility.

The probability formula is defined as the possibility of an event to happen is equal to the ratio of the number of favourableoutcomes and the total number of outcomes.

Probability of event to happen is

  $P\left(E\right)=\frac{Numberoffavourableoutcomes}{TotalNumberofoutcomes}$

  $P\left(Pointwillbeinshadedregion\right)=\frac{Areaofshadedregion}{Areaoflargercircle}=\frac{33\pi }{49\pi }=\frac{33}{49}=67.34\%$