In the adjoining figure, the inside perimeter of a running track with semi-circular ends and straight parallel sides is 312 m. The length of the straight portion of the track is 90 m. If the track has a uniform width of 2 m throughout, find its area.

Given,

Inside perimeter of the track = 312 m

Straight portions = 90 m each

Width of track = 2 m

Let inner radius be 'r' and outer radius be 'R'.

Inside perimeter consists of :

Two straights lengths = 90 + 90 = 180

Two semicircles = one full circle

So,

⇒ 180 + 2πr = 312

⇒ 2πr = 312 - 180

⇒ 2πr = 132

⇒ 2 × $\dfrac{22}{7}$ × r = 132

⇒ r = $\dfrac{132 \times 7}{2 \times 22}$

⇒ r = 21 m.

Outer radius (R) = r + 2

= 21 + 2 = 23 m.

Area of straight rectangular parts:

Area = 2 × (90 × 2)

= 360 m².

Area of curved part :

Area = π(R² - r²)

= π(23² - 21²)

= π(529 - 441)

= 88π

= 88 × $\dfrac{22}{7}$ = 276.57 m².

Total area of track = 360 + 276.57 = 636.57 m².

Hence, area of track = 636.57 m².