The inside perimeter of a running track (shown in Fig. 20.24) is 400 m. The length of each of the straight portion is 90 m

Question

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

Answer 1

Perimeter of the inner track = 2× Length of rectangle + perimeter of two semi-circular ends

Perimeter of the inner track = length + length + 2πr

400 = 90 + 90 + \(\frac{2\times 22\times r}{7}\)

\(\frac{2\times 22\times r}{7}\) = 400 - 180

r = 35 m

Hence, the radius of inner circle = 35 m

Radius of outer track = radius of inner track + width of the track

Radius of outer track = 35 + 14 = 49 m

Length of outer track = 2 × Length of rectangle + perimeter of two outer semi-circular ends

Length of outer track = 2 × 90 + 2πr

Length of outer track = 2 x 90 + 2πr

Length of outer track = 180 + \(\frac{2\times 22\times 49}{7}\)

Length of outer track = 488 m

Area of inner track = area of inner rectangle + area of two inner semi-circles

Area of inner track = length× breadth + πr²

Area of inner track = 90 x 70 + \(\frac{22\times 35\times 35}{7}\)

Area of inner track = 6300 + 3850

Area of inner track = 10150 m²

Area of outer track = area of outer rectangle + area of two outer semi-circles

Breadth of outer track = 35 + 35 +14 + 14 = 98 m

Area of outer track = length× breadth + πr²

Area of outer track = 90 x 98 + \(\frac{22\times 49\times 49}{7}\)

Area of outer track = 8820 + 7546

Area of outer track = 16366 m²

Area of path = area of outer track – area of inner track

Area of path = 16366 – 10150 = 6216 m²