Chapter 11.6, Problem 16PSB

a.

To determine

To find: The total shaded area.

Answer to Problem 16PSB

The totalshaded area is $1178c{m}^2$ .

Explanation of Solution

Given information:

The bull’s eye is of radius 5 cm and each band is 5 cm wide.

Formula used:

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

r = radius of a circle.

  $A_{ANNULUS}=\pi (R^2-r^2)$

R = radius of outer circle.

r = radius of inner circle.

Calculation:

  

Large dark band $=\pi (25^2-20^2)\approx 706.9c{m}^2$

Medium dark band $=\pi (15^2-10^2)\approx 392.7c{m}^2$

Bulls eye $=\pi {(5)}^2\approx 78.5c{m}^2$

  $\begin{array}{l}{A}_{shadedarea}=706.9+392.7+78.5\\ A_{shadedarea}=1178c{m}^2\end{array}$

b.

To determine

To calculate:The area of unshaded bands.

Answer to Problem 16PSB

The area of unshaded band is $785c{m}^2$ .

Explanation of Solution

Given information:

The bull’s eye is of radius 5 cm and each band is 5 cm wide.

Formula used:

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

r = radius of a circle.

  $A_{ANNULUS}=\pi (R^2-r^2)$

R = radius of outer circle.

r = radius of inner circle.

Calculation:

  

The radius of the entire figure is 25 cm.Each subsequent band has a radius 5 cm less than the previous band.The areas are as follows.

Large light band $=\pi (20^2-15^2)\approx 549.8c{m}^2$

Small light band $=\pi (10^2-5^2)\approx 235.6c{m}^2$

Bulls eye $=\pi {(5)}^2\approx 78.5c{m}^2$

  $\begin{array}{l}{A}_{unshadedarea}=706.9+392.7+78.5\\ A_{unshadedarea}=785c{m}^2\end{array}$

c.

To determine

To calculate: The probability to get a bull’s eye when we hit the target.

Answer to Problem 16PSB

The probability to get a bull’s eye when we hit the target is $\frac{1}{25}$ .

Explanation of Solution

Given information:

The bull’s eye is of radius 5 cm and each band is 5 cm wide.

Formula used:

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

r = radius of a circle.

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

Calculation:

  

The radius of the entire figure is 25 cm.Each subsequent band has a radius 5 cm less than the previous band.The areas are as follows.

Bulls eye area $=\pi {(5)}^2=25\pi$

Targeted area $=\pi {(25)}^2=625\pi$

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 favorable outcomes and the total number of outcomes.

Probability of event to happen is

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

  $P\left(Bullseyewhenwehitthetarget\right)=\frac{Areaofbullseye}{Areaoftarget}=\frac{25\pi }{625\pi }=\frac{1}{25}=4\%$