1)A wooden drum closed at the top and bottom has a top radius of 28cm and bottom radius 21cm,its height is 90cm.The drum is covered with leather excluding the top and bottom. Calculate;
a)The volume of the drum in litres
b)The area of the leather covering the drum excluding the top and bottom
a) To calculate the volume of the drum, we can use the formula for the volume of a cylinder: V = π r^2 h, where r is the radius and h is the height.
The radius of the top is 28 cm and the radius of the bottom is 21 cm. The height is 90 cm.
So, the volume of the drum is:
V = π (28^2 + 21^2) 90
V = π (784 + 441) 90
V = π (1225) 90
V ≈ 341,243 cm^3
To convert the volume to liters, we divide by 1000:
V ≈ 341,243 / 1000
V ≈ 341.243 liters
The volume of the drum is approximately 341.243 liters.
b) To calculate the area of the leather covering the drum, excluding the top and bottom, we need to find the lateral surface area of the drum.
The lateral surface area of a cylinder can be calculated using the formula: A = 2πrh, where r is the radius and h is the height.
The radius of the top is 28 cm and the radius of the bottom is 21 cm. The height is 90 cm.
So, the area of the leather covering the drum is:
A = 2π [(28 + 21) / 2] (90)
A = 2π (49 / 2) (90)
A = π (49) (90)
A ≈ 13,865.46 cm^2
The area of the leather covering the drum, excluding the top and bottom, is approximately 13,865.46 cm^2.
a) To calculate the volume of the drum in liters, we can use the formula for the volume of a cylinder:
V = πr^2h
Where V is the volume, r is the radius, and h is the height. However, we need to convert the measurements to meters before we can calculate the volume in liters.
Given:
Top radius (r1) = 28 cm = 0.28 m
Bottom radius (r2) = 21 cm = 0.21 m
Height (h) = 90 cm = 0.90 m
V = π × (r1^2 + r1 × r2 + r2^2) × h
= π × (0.28^2 + 0.28 × 0.21 + 0.21^2) × 0.90
≈ 2.41 m^3
To convert the volume to liters, multiply by 1000 (since there are 1000 liters in a cubic meter):
Volume in liters ≈ 2.41 × 1000
= 2410 liters
Therefore, the volume of the drum is approximately 2410 liters.
b) To calculate the area of the leather covering the drum, excluding the top and bottom, we need to find the surface area of the curved part of the drum.
The surface area of a cylinder (excluding the top and bottom) is given by the formula:
A = 2πrh
Given:
Top radius (r1) = 28 cm = 0.28 m
Bottom radius (r2) = 21 cm = 0.21 m
Height (h) = 90 cm = 0.90 m
A = 2π × (r1 + r2) × h
= 2π × (0.28 + 0.21) × 0.90
≈ 3.82 m^2
Therefore, the area of the leather covering the drum, excluding the top and bottom, is approximately 3.82 square meters.