Question

The figure shows a running track around a grassed enclosure which consists of rectangle PQRS with a surrounding semicircular region at each end. PQ = 200 m, PS = 70 m. Calculate

1. the area of the grassed enclosure in m².
2. the outer perimeter of the track if the width of the track is 7 m. `["Take"  π = 22/7]`

Answer 1

Given:

• PQ = 200 m   ...(Length of rectangle)
• PS = 70 m   ...(Breadth of rectangle)
• The figure consists of:
  • Rectangle PQRS
  • Two semicircles at both ends of the rectangle, forming a full circle
• Width of the running track = 7 m
• Use `π = 22/7`

i. Area of the grassed enclosure

The grassed area = rectangle + two semicircles (i.e., a full circle)

Area of rectangle = l × b

= 200 × 70

= 14,000 m²

Radius of each semicircle = `70/2` = 35 m

So, area of the full circle:

`πr^2 = 22/7 xx 35^2`

= `22/7 xx 1225`

= 3850 m²

Total grass area:

= Rectangle + Circle

= 14,000 + 3,850

= 17,850 m²

ii. Outer perimeter of the track

Now we calculate the outer boundary of the running track:

The track is 7 m wide throughout.

New width including track = 70 + 2 × 7 = 84 m

So new radius of semicircles = `84/2` = 42 m

New length of the rectangle (Remains unchanged) = 200 m

Perimeter = 2 straight sides + full circle

Straight sides (Lengths of rectangle) = 2 × 200 = 400 m

Curved part (Full circle) = 2πr

= `2 xx 22/7 xx 42`

= 264 m

Total outer perimeter:

= 400 + 264

= 664 m