Chapter 4.9, Problem 12P

To determine

To calculate: Find a formula for the area of the track in terms of $\pi$ and $r$ .

  

Answer to Problem 12P

Area of track is $8\pi r+16\pi +16r^2+64r-16r^2$

Explanation of Solution

Given information: Width of running track is 4 m and at each end of the soccer field the track is a semi circle with inner radius $r$ .

Calculation:

First we calculate the area of full region, here radius of bigger semicircle is $r+4$

Here width of rectangle is 2*radius $=2\left(r+4\right)$ , Hence length will be $2\left(2r\right)=4r$

so Area is

Area of soccer field = Area of rectangle +2(Area of semicircle)

  $A=2\left(\frac{1}{2}\pi {\left(r+4\right)}^2\right)+2\left(2\left(r+4\right)\right)\left(4r\right)$

  $A=\pi {\left(r+4\right)}^2+16r\left(r+4\right)$

Now we find the Area soccer field, here radius of semi-circles is $r$ , so width of smaller rectangle is $2r$ and its length is $2\left(2r\right)$

  $=2\left(\frac{1}{2}\pi r^2\right)$

Hence Area of soccer field is

  $=2\left(\frac{1}{2}\pi r^2\right)+2\left(2r\right)\left(4r\right)$

  $=\pi r^2+16r^2$

Area of track = Area of full region - Area of soccer field

  $=\left(\pi {\left(r+4\right)}^2+16r\left(r+4\right)\right)-\left(\pi r^2+16r^2\right)$

  $=\pi \left(r^2+8r+16\right)+16r^2+64r-\pi r^2-16r^2$

  $=8\pi r+16\pi +16r^2+64r-16r^2$

Conclusion:Area of track is $8\pi r+16\pi +16r^2+64r-16r^2$ .