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Please answer 10,11,12

Please answer 10,11,12-example-1
User Hbinduni
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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!

User Cristiano Ghersi
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