If you are given a rectangular canvas of 1.5 m in width, what length of this canvas would you require to make a conical tent that is 48 m in diameter and 7 m in height? Note that 10% of the canvas is used (wasted) in folds and stitchings.
Also, find the cost of the canvas at the rate of ₹24 per meter.

Given,

Diameter of conical tent = 48 m

Radius of conical tent (r) = $\dfrac{48}{2}$ = 24 m

Height of conical tent (h) = 7 m

Let l be the slant height of the conical tent.

By formula,

$\Rightarrow l = \sqrt{r^2 + h^2} \\[1em] \Rightarrow l = \sqrt{(24)^2 + 7^2} \\[1em] \Rightarrow l = \sqrt{576 + 49} \\[1em] \Rightarrow l = \sqrt{625} \\[1em] \Rightarrow l = 25 \text{ m}.$

By formula,

Surface area of conical tent (S) = πrl

$S = \dfrac{22}{7} \times 24 \times 25 \\[1em] = \dfrac{13200}{7} \text{ m}^2.$

Given,

10% of the canvas is used in folds and stitching.

Thus, 90% of the total canvas area is used for making the tent.

$\Rightarrow \dfrac{90}{100} \times \text{ Total canvas area} = \text{Surface area of conical tent} \\[1em] \Rightarrow \dfrac{90}{100} \times \text{ Total canvas area} = \dfrac{13200}{7} \\[1em] \Rightarrow \text{ Total canvas area} = \dfrac{13200}{7} \times \dfrac{100}{90} \\[1em] \Rightarrow \text{ Total canvas area} = \dfrac{132000}{63} \text{ m}^2.$

Thus,

⇒ Total canvas area = Length of canvas × Width of canvas

⇒ $\dfrac{132000}{63}$ = Length of canvas × 1.5

⇒ Length of canvas = $\dfrac{132000}{63 \times 1.5} = \dfrac{132000}{94.5}$ = 1396.83 m

Cost of canvas = Rate per m × Length

= 24 × 1396.83

= ₹ 33,523.92

Hence, length of canvas required = 1396.83 m and cost of canvas = ₹ 33,523.92