Steve S. answered • 03/31/14

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Tutoring in Precalculus, Trig, and Differential Calculus

Dimensions of paper: x by y

2x + 2y = 100

x + y = 50

Volume of cylinder: V = Bh = pi r^2 h

Let x/2 = r and y = h = 50 – x:

V = pi (x/2)^2 (50 – x)

Domain: 0 < x < 50

V = pi/4 x^2 (50 – x)

Extrema when V’ = 0

V’ = pi/4 (2x(50 – x) – x^2) = 0

100x – 2x^2 – x^2 = 0

100x – 3x^2 = 0

x(100 – 3x) = 0

Domain excludes x = 0

100 – 3x = 0

x = 100/3 = 33+1/3 cm

y = 50 – 33 – 1/3 = 16+2/3 cm