A tent is of the shape of a right circular cylinder upto height of 3 meters and then becomes a right circular cone with a maximum height of 13.5 meters above the ground. Calculate the cost of painting the inner surface of the tent at ₹ 4 per sq. meter, if the radius of the base is 14 meters.

Given,

Height of cylindrical portion (h) = 3 m

Height of conical part (H) = Maximum height of tent (above ground) - Height of cylindrical portion = 13.5 - 3 = 10.5 m.

Radius of base of cylindrical portion = Radius of conical portion = r = 14 m.

⇒ Curved surface area of tent = Curved surface area of cone + Curved surface area of cylinder

⇒ Curved surface area of tent (C) = πrl + 2πrh …….(1)

We know that,

$l = \sqrt{r^2 + H^2} \\[1em] = \sqrt{14^2 + (10.5)^2} \\[1em] = \sqrt{196 + 110.25} \\[1em] = \sqrt{306.25} \\[1em] = 17.5 \text{ m}.$

Substituting values in equation (1), we get :

$\Rightarrow C = \dfrac{22}{7} \times 14 \times 17.5 + 2 \times \dfrac{22}{7} \times 14 \times 3 \\[1em] = 22 \times 2 \times 17.5 + 2 \times 22 \times 2 \times 3 \\[1em] = 770 + 264 \\[1em] = 1034 \text{ m}^2.$

Given,

Cost of painting the inner surface of the tent at ₹ 4 per sq. meter.

Total cost = Curved surface area of tent × 4

= 1034 × 4 = ₹ 4136.

Hence, cost of painting the inner surface = ₹ 4136.