Mathematics

In the following diagram a rectangular platform with a semicircular end on one side is 22 meters long from one end to the other end. If the length of the half circumference is 11 meters, find the cost of constructing the platform, 1.5 meters high at the rate of ₹ 4 per cubic meters.

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Given,

Length of the platform (L) = 22 m

Half circumference = 11 m

∴ πr = 11

⇒ r = $\dfrac{11 \times 7}{22} = \dfrac{7}{2}$ = 3.5 m

From figure,

Breadth of rectangular part (b) = 2r = 2 × 3.5 = 7 m.

Length of rectangular part (l) = Total length (L) - radius (r)

= 22 - 3.5 = 18.5 m.

Area of cross-section = lb + $\dfrac{1}{2}πr^2$

$= 18.5 \times 7 + \dfrac{1}{2} \times \dfrac{22}{7} \times (3.5)^2 \\[1em] = 129.5 + 19.25 \\[1em] = 148.75 \text{ m}^2.$

Given, Height of the platform (h) = 1.5 m

Volume = Area of cross-section x height

= 148.75 x 1.5 = 223.125 m³.

Rate of construction = ₹ 4 per m³

Total expenditure = ₹ 4 x 223.125 = ₹ 892.50.

Hence, cost of constructing the platform = ₹ 892.50.

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