need help with #7 and 8 iPad 2:43 PM 82%-) webassign.net hw 07 Practice Another

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need help with #7 and 8 iPad 2:43 PM 82%-) webassign.net hw 07 Practice Another Version 7" + 1 points LarCalc10 3.7-033.MI. My Notes A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 18 cubic centimeters. Find the radius of the cylinder that produces the minimum surface area. (Round your answer to three decimal places.) Need Help? ReadtatrChat About 8. 0/1 points I Previous Answers LarCalc10 3.7.038. My Notes Two factories are located at the coordinates (-x, 0) and (x, 0), and their power supply is at (0, h) (see figure). Find y such that the total length of power line from the power supply to the factories is a minimum. (0, h) (x, 0) Need Help? Read" llenatAboutRI

Explanation / Answer

solution

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r = the radius of thecylinder and the hemispheres
h = the height of the cylinder
Note that the two hemispheres are equivalent to a sphere.
The volume of the cylinder is:
V = pr2h
The volume of the hemispheres is:
V = ( 4 / 3 )pr3
The surface area of the cylinder ( excluding its base and top )is:
A = 2prh
The surface area of the hemispheres is:
A = 4pr2
The volume of the solid is:
V = The volume of the cylinder + The volume of the hemispheres
    = pr2h + ( 4 / 3 )pr3
V = 18 cm3

=> pr2h + ( 4 / 3)pr3 = 18

=> pr2h = 18 - ( 4 / 3)pr3 =>
h = 18 / ( pr2 ) - ( 4 / 3)r
The surface area of the solid is:
A = The surface area of the cylinder (excluding its base and top ) + The surface areaof the hemispheres
    = 2prh + 4pr2
    = 2pr[ 18/ (pr2 ) - ( 4 / 3 )r ] + 4pr2
    = [ 36 / r - ( 8 /3 )pr2 ] + 4pr2
    = ( 4 / 3 )pr2 + 36 / r
Minimize A:
A' = ( 8 / 3 )pr - 36 /r2
A' = 0

=>( 8 / 3 )pr - 36 / r2 = 0
Multiply both sides of the equation by r2
( 8 / 3 )pr3 - 36 = 0

=>( 8 / 3 )pr3 = 36

=>
r3 = 36[ 3 /( 8p ) ] =>

r3 = 27 /2 p =>

r = ( 27/ 2p)1 / 3

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