Calculus Help- Max. volume of a cylinder vs. Min. Surface Area?

Question

I keep getting half way there, but somewhere in my solution, I keep getting messed up. Any help is more than welcome! Thanks in advance. BTW, this is for a Calc 1 course.

Paul's Porters is looking to create a 2.5 gallon cylindrical jug to distrubute his brewed bliss to the populace. In order to sell his jugs at the lowest cost possible, Paul wished to design a cylindrical jug that uses the minimal amount of materials. What are the dimenstions of the cliindrical jug that minimize surface area (materials used)?

Hints:
a) 1 gallon = 230 cubic inches
b) The volume of a cylinder is V = π r² h
c) The surface area of a cylinder is SA = 2πr² + 2π rh

Answer 1

Imagine if the cylinder were a cube: the 2πrh portion accounts for 4 sides, and the 2πr² (the base) accounts for 2 sides. Thus, to make your cylinder most like a cube, you need to set the base to be half the size of the tube.
2πrh = 2 * 2πr²
rh = 2r²
h = 2r

Now, redefine h as 2r for your volume and SA formulas to find the most compact surface area:
V = πr²h = 2πr^3
575 = 2πr^3
r^3 = 91.514
r = 4.506 cm, so h = 9.012 cm. (r and h solution!)

SA = 2πr² + 2πrh
SA = 2π(r² + rh) = 2π(r² + 2r²) = 6πr²
SA = 6π(4.506)² = 382.722 sq. cm (SA solution!)

Check: set r to 4.5 and 4.51.
V = πr²h
h = V/πr² = 575 / (π * 20.25) = 9.038 cm.
SA = 2πr² + 2πrh
SA = 2π(4.5)² + 2π(4.5 * 9.038)
SA = 127.235 + 255.543 = 382.778 (greater)

V = πr²h
h = V/πr² = 575 / (π * 20.3401) = 8.998 cm.
SA = 2πr² + 2πrh
SA = 2π(4.51)² + 2π(4.51 * 8.998)
SA = 127.801 + 254.978 = 382.779 (greater)

Answer 2

volume of a cylinder: R^2 * H * pi SA of a cylinder: 2 * pi * R^2 + H * 2 * pi * R SA / V = 2 * R * pi ( R + H) / (R^2 * H * pi) 2 (R + H) / (R * H) so which you prefer to discover the minimum fee for this function, you should take the 1st spinoff and discover while f ' (H) = 0 ( (R * H) * d/dH (2 ( R + H )) - 2 ( R + H ) * d/dH (R * H) ) / (R^2 * H^2) -2R ( R + H ) / (R^2 + H^2) 0 = -2R (R + H) 0 = -2R^2 - 2RH while R = H, you'll have the optimal top. that'd be 6 cm. a) 6 cm b) 60mm/2mm = 30 CDs