Chapter 11.5, Problem 23WE

(a)

To determine

To Find: The area of bull eye and each ring of target.

Answer to Problem 23WE

  $3\pi ,5\pi ,7\pi$

Explanation of Solution

Given: A target consist of four concentric circles of radii $1,2,3,4$ .

Given: $radius=1,2,3,4$

Area of circles with radius $1,2$ .

  $\begin{array}{l}{A}_1=\pi r_1{}^2\\ $ A_1=\pi {(1)}^2=\pi $$ A_2=\pi {(2)}^2=4\pi $\end{array}$

Now the area of the first ring is the difference of area of second circle and bull eye.

Thus

  $\begin{array}{l}A=4\pi -\pi \\ A=3\pi \end{array}$

i.e. the area of the first ring is $3\pi$ .

Now

Area of circles with radius $3,4$ .

  $\begin{array}{l}{A}_1=\pi r_1{}^2\\ $ A_3=\pi {(3)}^2=9\pi $\end{array}$

The area of the second ring is the difference of area of circle with radius three and two.

  $A=9\pi -4\pi =5\pi$

Area of circle with radius $4$

  $\begin{array}{l}A=\pi r_1{}^2\\ $ A_4=\pi {(4)}^2=16\pi $\end{array}$

Area of third ring,

  $A=16\pi -9\pi =7\pi$

Hence, the area of the rings are $3\pi ,5\pi ,7\pi$

(b)

To determine

To Find: The area of $nth$ ring if the target consist of $n$ rings and bull eye.

Answer to Problem 23WE

  $A=(2n+1)\pi$

Explanation of Solution

Given: A target consist of four concentric circles of radius $1,2,3,4$ .

Area of circle with radius $(n+1)$ is

  $A=\pi {(n+1)}^2$

Area of circle with radius $n$ is

  $A=\pi {(n)}^2$

Now area of $nth$ ring is

  $\begin{array}{l}$ A=\pi {(n+1)}^2-\pi n^2$A=(2n+1)\pi \end{array}$