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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) §3.2 Maximum and Minimum Problems, #49 · Calculus · Optimization

Including top and bottom, the cylinder has area …A=2\pi rh+2\pi r^2=2\pi rH(1-(r/R))+2\pi…

Including top and bottom, the cylinder has area
$A = 2 π r h + 2 π r^{2} = 2 π r H (1 - (r / R)) + 2 π r^{2} .$
Maximize $A$ when $H > R$ . Maximize $A$ when $R > H$ .

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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §3.2 Maximum and Minimum Problems, #49

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