Find the minimum length in cm and correct to nearest whole number of the thin metal sheet required to make a hollow and closed cylindrical box of diameter 20 cm and height 35 cm. Given that the width of the metal sheet is 1 m. Also, find the cost of the sheet at the rate of Rs. 56 per m.

Find the area of metal sheet required, if 10% of it is wasted in cutting, overlapping, etc.

Given,

Height of the cylinder box (h) = 35 cm

Base radius of the cylinder box (r) = $\dfrac{20}{2}$ = 10 cm

Width of metal sheet = 1m = 100 cm

Area of metal sheet required = Total surface area of the box

⇒ Length x width = 2πr(r + h)

⇒ Length x 100 = 2 x $\dfrac{22}{7}$ x 10(10 + 35)

⇒ Length x 100 = 2 x $\dfrac{22}{7}$ x 10 x 45

⇒ Length = $\dfrac{2 \times 22 \times 10 \times 45}{7 \times 100}$ = 28.28 cm ≈ 28 cm (correcting to the nearest whole number)

Thus,

Area of metal sheet = length x width = 28 x 100 = 2800 cm² = $\dfrac{2800}{100 \times 100}$ = 0.28 m².

So, the cost of the sheet at the rate of ₹ 56 per m² = ₹ (56 x 0.28) = ₹ 15.68

Let the total sheet required be x.

Then, x - 10 % of x = 2800 cm

⇒ $x - \dfrac{10}{100} \times x$ = 2800

⇒ $\dfrac{100x - 10x}{100}$ = 2800

⇒ $\dfrac{90x}{100}$ = 2800

⇒ $\dfrac{9x}{10}$ = 2800

⇒ $x = \dfrac{2800 \times 10}{9}$

⇒ x = 3111.11 cm² ≈ 3111 cm² (correcting to the nearest whole number)

Hence, length = 28 cm, cost of sheet = ₹15.68 and area of sheet required = 3111 cm².