Chapter 8.4, Problem 55PPS

a.

To determine

To determine the area of the outer circle.

Answer to Problem 55PPS

  $18\pi r+81\pi$or $56.548r+254.469$

Explanation of Solution

Given:

  

Radius of inner circle is $r$ feet and radius of outer circle is $9$ feet longer than the inner circle.

Formula used:

Area of circle $=\pi \times {(\text{radius)}}^2$

  ${(a+b)}^2=a^2+b^2+2ab$

Calculation:

First find the area of the inner circle,

Area of inner circle $=\pi \times {(\text{radius)}}^2$

        $=\pi \times {(r)}^2$

        $=\pi r^2$

Now, finding out the area of outer circle,

Area of circle $=\pi \times {(\text{radius)}}^2$

      $=\pi \times {(r+\text{9)}}^2$

      $=\pi \times (r^2+2\times r\times 9+81)\left[\because {(a+b)}^2=a^2+b^2+2ab\right]$

      $=\pi {\text{r}}^2+18\pi r+81\pi$

Now this is the area of outer circle including the inner circle.

Removing the inner circle area from this total outer area circle can give the outer circle area,

Area of the outer circle $=\pi r^2+18\pi r+81\pi -\pi r^2$

          $=18\pi r+81\pi$ or $56.548r+254.469$

Conclusion:

Therefore, area of the outer circle is $18\pi r+81\pi$ or $56.548r+\mathrm{254.469.}$

b.

To determine

To determine the expression of the area of the square outside the larger circle.

Answer to Problem 55PPS

  $1189.53-\pi r^2-56.548$

Explanation of Solution

Given:

  

The square of the wrestling mat is 38 foot sides, Radius of inner circle is $r$ feet and radius of outer circle is $9$ feet longer than the inner circle.

Formula used:

Area of square $={(\text{side)}}^2$

Calculation:

Now, from the above two area of smaller circle including the outer yellow circle is $\pi {\text{r}}^2+18\pi r+81\pi .$

For the area of outer portion of the square outside the circle, first take the area of whole square.

Whose side is $38$ feet.

So,

Area of square $={(38\text{)}}^2$

       $=1444$ square feet

Now,

Area of square portion outside the outer yellow circle $=$ Area of square $-$ area of outer yellow circle including smaller circle.

  $=1444$ square ft $-$ $(\pi {\text{r}}^2+18\pi r+81\pi )$ square feet

  $=1189.53-\pi r^2-56.548$ square feet

Conclusion:

Therefore, area of square portion outside the outer yellow circle is $1189.53-\pi r^2-56.548$ square feet.