Final answer:
The minimum length of the thin metal sheet required to make a hollow and closed cylindrical box is approximately 9π cm. The cost of the metal sheet, at a rate of Rs 56 per sq m, is approximately 4.928π Rs.
Step-by-step explanation:
To find the minimum length of the thin metal sheet required to make a hollow and closed cylindrical box, we need to calculate the outer surface area of the box. The outer surface area of a cylinder is given by the formula:
A = 2πrh + πr^2
Where r is the radius of the cylinder and h is the height. In this case, the diameter is given as 20 cm, so the radius is 10 cm. And the height is given as 35 cm. Substituting these values into the formula, we get:
A = 2π(10)(35) + π(10)^2 = 700π + 100π = 800π
Now, we need to find the area of the metal sheet required. Since 10% of it is wasted, we need to multiply the outer surface area by 1.1:
Area of metal sheet = 800π * 1.1 = 880π cm²
Next, we convert the width of the metal sheet from meters to centimeters. 1 meter = 100 centimeters, so the width is 100 cm. Now we can calculate the length of the metal sheet required:
Length of metal sheet = Area of metal sheet / Width = 880π / 100 = 8.8π cm
Finally, we can round the length to the nearest whole number:
Length of metal sheet (rounded) = 9π cm
Now, to find the cost of the metal sheet, we need to calculate the area of the metal sheet and then multiply it by the rate per square meter. The rate is given as Rs 56 per sq m:
Area of metal sheet = 880π cm² = 880π / 10000 m² (since 1 m² = 10000 cm²) = 0.088π m²
Cost of metal sheet = Area of metal sheet * Rate = 0.088π * 56 = 4.928π Rs
Therefore, the minimum cost of the metal sheet is approximately 4.928π Rs.