The inside perimeter of a running track shown in the figure is 400 m. The length of each of the straight portions is 90 m, and the ends are semicircles. If the track is 14 m wide everywhere, find the area of the track. Also, find the length of the outer boundary of the track.

Question

Philoid · Accepted Answer

Consider the figure as a combination of two semicircles on the ends of the rectangle

Let the length of rectangle be ‘L’ m and breadth be ‘B’ cm

Given L = 90 m, W = 14 m

Perimeter of running track = 400 m

Perimeter of inside of running track = 2L + Arc of two semicircles → eqn1

Arc length of a semicircle = πr where r = radius

⇒ 400 = (2×90) + (2×πr) (putting values in eqn1)

⇒ 400 = 180 + 2πr

⇒ 400 – 180 = 2πr

⇒ 220 = 2πr

∴ r = 35 m

Area of inner of running track = Area of rectangle + 2×area of semicircles → eqn2

Area of rectangle = L×B

Here B = 2r

B = 70 m

⇒ Area of inner rectangle = 90×70

∴ Area of inner rectangle = 6300 m²→ eqn3

⇒ Area of inner semicircles = 1225π

⇒ Area of inner semicircle = 175×22

∴ Area of inner semicircle = 3850 m²→ eqn4

Area of inner of running track = 6300 + 3850 (from3&4)

∴ Area of inner of running track = 10150 m²

Now radius of semicircles of outer of the running track = R = r + W

⇒ R = 35 + 14

∴ R = 49 m

⇒ Area of outer semicircles = 2401π

⇒ Area of outer semicircles = 343×22

∴ Area of outer semicircle = 7546 m²→ eqn5

Breadth of outer running track = B’ = 2R

⇒ B’ = 2×49

∴ B’ = 98 m

Area of outer rectangle = L×B’

⇒ Area of outer rectangle = 90×98

∴ Area of outer rectangle = 8820 m²→ eqn6

Area of entire ground = 8820 + 7546 (from 5&6)

∴ Area of entire ground = 16366 m²

Area of running track = Area of entire ground – Area of inner ground

⇒ Area of running track = 16366 – 10150

∴ Area of running track = 6216 m²

Perimeter of outer boundary = 2L + Arc of outer semicircles

Arc length of an outer semicircle = πR, where R = outer radius

⇒ Perimeter of outer boundary = 180 + (2×22×7)

⇒ Perimeter of outer boundary = 180 + 308

∴ Perimeter of outer boundary = 488 m

Area of running track is 6216 m² and perimeter of outer boundary is 488 m.