Question

The inside perimeter of a running track (shown Fig.) is $400$ m. The length of each of the straight portion is $90$ m and the ends are semi-circles. If the track is everywhere $14$ m wide,find the length of outer track. Also find the area of track.

Answer 1

The inside perimeter of the track $=400m$

The total length of the two straight portions $=90+90=180m$

Therefore the length of the remaining portion $=400-180=220m$

Circumference of the two remaining semi-circular portions $=\pi r+\pi r=2\pi r$. ${}^{\prime }{r}^{\prime }$ is radius

$\Rightarrow$ $2\pi r=220$

$\Rightarrow$ $2\times \frac{22}{7}\times r=220$

$\therefore$ $r=35m$

So, the radius of the circular portion of the outer running running track $=35m+14m=49m$

Area of the track = Area of the two rectangles of dimensions + The area of the circular rings.

$\Rightarrow$ Area of track $=2\times 90\times 14+\frac{22}{7}\times \left[\right(49{)}^2-\left(35{)}^2\right]$

$\Rightarrow$ Area of track $=2520+\frac{22}{7}\times (2401-1225)$

$\Rightarrow$ Area of track $=2520+\frac{22\times 1176}{7}$

$\Rightarrow$ Area of track $=2520+3696$

$\therefore$ Area of track $=6216m^2$

$\Rightarrow$ Length of the outer running track $=2\times 90+2\times \frac{22}{7}\times 49$

$\Rightarrow$ Length of the outer running track $=180+308$

$\therefore$ Length of the outer running track $=488m$