Question

The cross-section of a tunnel perpendicular to its length is an isosceles trapezium as shown in the figure. If AB = 8 m, DC = 7 m, AD = BC and DM = 1.2 m and the tunnel is 100 m long, then calculate:

1. the cost of painting the internal surface of the tunnel (excluding the floor) at the rate of ₹ 6 per m².
2. the cost of paving the floor at the rate of ₹ 20 per m².
3. the cubic content of the tunnel.

Answer 1

Given:

Cross-section is an isosceles trapezium with AB = 8 m (bottom), DC = 7 m (top), AD = BC, DM = height = 1.2 m. Tunnel length = 100 m.

Use: Volume = (area of cross-section) × length and lateral surface (excluding floor) = (sum of top + 2 side lengths) × length.

Step-wise calculation:

1. Find the slant side AD and BC.

Horizontal offset on each side

= `(AB - DC)/2`

= `(8 - 7)/2`

= 0.5 m

`AD = sqrt((0.5)^2 + (1.2)^2)`

= `sqrt(0.25 + 1.44)`

= `sqrt(1.69)`

= 1.3 m

2. Internal surface to be painted (excluding floor).

Per cross-section (top + 2 sides)

= DC + AD + BC

= 7 + 1.3 + 1.3

= 9.6 m

Painted area

= (9.6 m) × (length 100 m) 

= 960 m²

Painting cost

= 960 × ₹ 6 

= ₹ 5,760

3. Floor paving

Floor area

= AB × length

= 8 × 100

= 800 m²

Paving cost

= 800 × ₹ 20 

= ₹ 16,000

4. Cubic content (volume).

Area of trapezium cross-section

= `1/2` × (AB + DC) × height

= 0.5 × (8 + 7) × 1.2

= 0.5 × 15 × 1.2

= 9.0 m²

Volume

= Area × Length 

= 9.0 × 100

= 900 m³

1. Cost of painting internal surface (excluding floor) = ₹ 5,760.
2. Cost of paving the floor = ₹ 16,000.
3. Cubic content (volume) of the tunnel = 900 m³.