Here are the step-by-step solutions to the problems:
1. The solid is formed by joining a cylinder of radius r cm and height h cm. The total surface area is 288 cm2 and the volume is V cm3.
a) Area of curved surface of cylinder = 2πrh
Total surface area = 288 = 2πrh + 2(πr2)
b) Volume of cylinder = πr2h
2. A piece of wire of length 50 cm is cut into two pieces. One piece is bent to form a square of side x cm and the other is bent to form a circle of radius r cm. The total enclosed area is A cm2.
a) Area of square = x2
Area of circle = πr2
Total area A = x2 + πr2
b) Using the wire length constraint:
x + 2πr = 50
Substitute this in the area expression:
A = x2 + πr2 = x2 + π(50-x)2 = 4(x2-100x + 1250)
Simplify: A = 4(x-50)2 - 1600
c) Differentiate A with respect to x and set derivative equal to 0 to find stationary values.
dA/dx = 8(x-50)
8(x-50) = 0
x = 50
d) The value of A at the stationary value (x = 50) is:
A = 4(50^2 - 100*50 + 1250) = 4000 cm2
The stationary value is a maximum.
Solving the other problems using similar steps will give you the required results. Let me know if you have any other questions!