Question

The cross-section of a tunnel is a square of side 7 m surmounted by a semi-circle as shown in the adjoining figure. The tunnel is 80 m long.

Calculate:

1. its volume,
2. the surface area of the tunnel (excluding the floor) and
3. its floor area.   

Answer 1

Side of square = 7 m

Radius of semicircle = `7/2` m  

Length of the tunnel = 80 m 

Area of cross-section of the front part = `a^2 + 1/2pir^2` 

= `7 xx 7 + 1/2 xx 22/7 xx 7/2 xx 7/2` 

= `49 + 77/4 m^2`  

= `(196 + 77)/4` 

= `273/4 m^2` 

i. Therefore, volume of tunnel = area × length 

= `273/4 xx 80`  

= 5460 m³

ii. Circumference of the front of tunnel  

= `2 xx 7 + 1/2 xx 2pir` 

= `14 + 22/7 xx 7/2` 

= 14 + 11 

= 25 m 

Therefore, surface area of the inner part of the tunnel 

= 25 × 80 

= 2000 m² 

iii. Area of floor = l × b

= 7 × 80

= 560 m²