Largest Possible Volume for a Cylinder...?

14
March

Using a 22in x 28in poster board, what is the largest possible cylinder you can make? Note that it must be a "true cylinder," i.e. the top and bottom circles are the same size. The pieces do not have to be whole, i.e. you can use scraps to add the to circumference of the barrel of the cylinder so that the circles can be made wider.

Pertinent Formulas:

Circumference of a Circle: 2r(pi)
Volume of a Cylinder: hr^2(pi)

A = 2� (r² + hr) .................. surface area is limited to (22x28) in²
h = 308/(� r) - r
dh/dr = -308/(� r²) - 1

V = � r² h
dV/dr = 2� r h + � r² dh/dr
= 2� r (308/(� r) - r) + � r² (-308/(� r²) - 1)
= 308 - 3� r²
dV²/dr² = -6� r

Find min/max by setting dV/dr = 0
3� r² = 308
r = 2√(77/(3� )) ............... dV²/dr² < 0, so it's a max

Answer: Max volume is (1232/3) √(77/(3� )) in² with r = 2√(77/(3� )) in and h = 4√(77/(3� )) in

Taylor, double check your calculations being mindful of the parentheses: 2√(77/(3� )) ≈ 5.72

77/(3� ) ≈ 8.17
√(8.17) ≈ 2.86
2 (2.86) = 5.72

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Largest Possible Volume for a Cylinder...?

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