Directions:

At an NCC camp, several tents were installed. Each tent is cylindrical to a height of 3 m and conical above it. The total height of the tent is 13.5 m and the radius of its base is 14 m.

Based on this information, answer the following questions:

61. The slant height of the conical portion of the tent is :

(a) 16.5 m
(b) 17.5 m
(c) 18.5 m
(d) 19.5 m

62. The cost of cloth required to make each tent at the rate of ₹ 80 per square meter is :

(a) ₹ 76560
(b) ₹ 80140
(c) ₹ 82720
(d) ₹ 85960

63. If each cadet requires 8 m² of floor space and there are 15 tents in all how many cadets can be accommodated in the camp?

(a) 960
(b) 1155
(c) 1320
(d) 1440

64. If a tent has maximum number of cadets that it can accommodate as calculated in the above questions, what is the volume of air available to each cadet to breathe?

(a) 48 m³
(b) 52 m³
(c) 55 m³
(d) 77 m³

61. Given,

Height of cylinder, h = 3 m

Total height of tent, T = 13.5 m

Height of cone, H = T - h = 13.5 - 3 = 10.5 m

Radius of base of cylinder = Radius of cone = r = 14 m

Slant height of cone be l m.

l² = r² + H²

⇒ l² = 14² + 10.5²

⇒ l² = 196 + 110.25

⇒ l² = 306.25

⇒ l = $\sqrt{306.25}$ = 17.5 m

Hence, Option (b) is the correct option.

62. Curved surface area of tent = Curved surface area of cone + Curved surface area of cylinder

= 2πrh + πrl

= πr(2h + l)

$= \dfrac{22}{7} \times 14 (2 \times 3 + 17.5) \\[1em] = 22 \times 2(6 + 17.5) \\[1em] = 44 \times 23.5 \\[1em] = 1034 \text{ m}^2.$

Given, cost of cloth required to make each tent is ₹ 80 per square meter.

⇒ Total cost = 80 × 1034 = ₹ 82720

Hence, Option (c) is the correct option.

63. The floor space of a tent is base area of cylinder.

∴ Area of base = πr²

= $\dfrac{22}{7}$ × 14 × 14

= 22 × 2 × 14

= 616 m²

Given each cadet requires 8 m² of floor space.

The number of cadets per tent = $\dfrac{\text{Area of base}}{\text{space required per cadet}} = \dfrac{616}{8}$ = 77 cadets.

Given, there are 15 tents.

∴ Total number of cadets = 77 × 15 = 1155 cadets.

Hence, Option (b) is the correct option.

64. Volume of air in each tent = Volume of air in cylinder + Volume of air in cone

$= π\text{r}^2\text{h} + \dfrac{1}{3}π\text{r}^2\text{H} \\[1em] = π\text{r}^2 (\text{h} + \dfrac{1}{3}\text{H}) \\[1em] = \dfrac{22}{7} \times 14^2 (3 + \dfrac{1}{3} \times 10.5) \\[1em] = \dfrac{22}{7} \times 196 (3 + 3.5) \\[1em] = 22 \times 28 \times 6.5 \\[1em] = 4004 \text{ m}^3.$

Volume of air available to each cadet to breathe = $\dfrac{\text{Volume of air per tent}}{\text{Cadets per tent}} = \dfrac{4004}{77}$ = 52 m³.

Hence, Option (b) is the correct option.