The radius of the cylinder that produces the minimum surface area is 1.34cm and this can be determined by using the formula area and volume of cylinder and hemisphere.
A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder.
The total volume of the solid is 10 cubic centimeters.
The volume of a cylinder is given by:
V = πr²h
The total volume of the two hemispheres is given by:
V = 2 x 2/3 πr³
Now, the total volume of the solid is given by:
V total = πr²h + 2 x 2/3πr³
Now, substitute the value of the total volume in the above expression and then solve for h.
10 = πr²h + 4/3πr³
h = 10/πr² + 4r/3
Now, the surface area of the curved surface is given by:
A = 2πrh
Now, the surface area of the two hemispheres is given by:
A' = 2 x 2πr²
A' = 4πr²
Now, the total area is given by:
A total = 2πrh + 4πr²
Now, substitute the value of 'h' in the above expression.
A total = 2πr(h = 10/πr² + 4r/3) + 4πr²
Simplify the above expression.
dA = -20/r + 4πr²/3
Now, differentiate the total area with respect to 'r'.
dA/dr = 20/r + 8πr/3
Now, equate the above expression to zero.
0 = -20/r² + 8πr/3
Simplify the above expression in order to determine the value of 'r'.
8πr³ = 60
r = 1.34 cm
Therefore, the radius of the cylinder that produces the minimum surface area is 1.34cm
To learn more about area of a region refer here
brainly.com/question/11952845
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