Mathematics

The cross-section of a railway tunnel is a rectangle 6 m broad and 8 m high surmounted by a semi-circle as shown in the figure. The tunnel is 35 m long. Find the cost of plastering the internal surface of the tunnel (excluding the floor) at the rate of ₹ 2.25 per m².

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Given,

Breadth of tunnel (b) = 6 m

Height of tunnel (h) = 8 m

Length of tunnel (l) = 35 m

Let radius of semi-circle be r meters.

From figure,

⇒ 2r = 6

⇒ r = 3 m.

Circumference of semi-circle = πr = $\dfrac{22}{7} \times 3 = \dfrac{66}{7}$ m.

Internal surface area of tunnel = Circumference of semi-circle × Length + Area of side interior rectangular walls

= πrl + hl + hl

= πrl + 2hl

= $\dfrac{66}{7} \times 35 + 2 \times 8 \times 35$

= 330 + 560

= 890 m².

Given,

Rate of plastering = ₹ 2.25 per m².

Total cost = 890 × ₹ 2.25 = ₹ 2002.50.

Hence, cost of plastering internal surface area of tunnel = ₹ 2002.50

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