Calculus question

Question

A solid is formed by attaching a hemisphere to each end of a cylinder. If the total volume is to be 120cm³, find the radius (in cm) of the cylinder that produces the minimum surface area. Express your answer correct to 2 decimal places. 

Help?!

Answer 1

 

V = 4/3*pi*r^3 + pi*r^2 = 120

 

Solving for h:  H = (120 - 4/3*pi*r^3)/(pi*r^2)  <--- label this equation ALPHA

 

SA = 2*pi*r*h + 4*pi*r^2

 

     = 2*pi*r*[ (120 - 4/3*pi*r^3)/(pi*r^2)] + 4*pi*r^2 <--- substitutes equation ALPHA for height H

 

    = (240 - 8/3*pi*r^3)/r + 4*pi*r^2 <--- cancels pi*r and distributes the 2

 

    = (240 - 8/3*pi*r^3 + 4*pi*r^3)/r   <--- common denominator of r

 

    = (240 - 8/3*pi*r^3  + 12/3*pi*r^3)/r  <---- 4 = 12/3 common denominator in the numerator

 

   = (240 + 4/3*pi*r^3)/r

 

 

Now, takes the derivative of the surface area function. Extrema found by setting the derivative equal 

to zero and solving for r:

 

  0 = SA'(r) = d(SA)/dR = [r(4*pi*r^2) - (240 + 4/3*pi*r^3)]/(r^2)  <--- quotient rule

 

                                  = (4*pi*r^3 - 240 - 4/3*pi*r^3)/(r^2) <-- distributes r and -1 which changes the signs

 

                                   = (8/3*pi*r^3 - 240)/(r^2)  <----   4 = 12/3 minus 4/3; combines like terms 

 

denominator cancels on solving for R

 

8/3*pi*r^3 - 240 = 0

 

pi*r^3 = 240*3/8 = 90

 

r^3 = 90/pi

 

r = cube-root(90/pi) = 3.06

 

Extrema of surface area = 240 + 4/3*pi*90/pi divided by cube root of (90/pi)

 

                      = 360 * (90/pi) ^(2/3) /  (90/pi) <--- rationalizes the denominator

 

                      = pi/4*(90/pi)^(2/3) <--- dividing fractions:KFC; 360 cancels 90, pi goes on top

 

                      = pi/4 * (90/pi)^(2/3)

 

                      = 7.353345....

 

The radius is highlighted in bold above

 

Answer 2

The surface area of the solid is

 

SA = 4πr² + 2πrh                        eq1

 

(4/3)πr³ + πr²h = 120                eq2

 

4πr³ + 3πr²h = 360                      eq2   ---> rewritten

 

 

Next, we get the surface area equation in terms of r by substituting eq2 into eq1.

 

h = (360 - 4πr³) / 3πr²

 

 

SA = 4πr² + 2πr[(360 - 4πr³) / 3πr²]

SA = 4πr² + 2πr[(120 / πr²) - (4/3)r]

SA = 4πr² + 240r^-1 - (8/3)πr²

SA = 240r^-1 + (4/3)πr²

 

Now we can take the first derivative of SA and set it equal to zero.

 

SA' = 0

 

-240r^-2 + (8/3)πr = 0

 

-8r^-2(30 - (1/3)πr³) = 0

 

30 - (1/3)πr³ = 0

 

(1/3)πr³ = 30

 

πr³ = 90

 

r = ³√(90/π)