Question

Minimum painting surface A metal cistern in the shape of a right circular cylinder with volume V = 50 m³ needs to be painted each year to reduce corrosion. The paint is applied only to surfaces exposed to the elements (the outside cylinder wall and the circular top). Find the dimensions r and h of the cylinder that minimize the area of the painted surfaces.

Accepted Answer

Start by identifying the formulas needed: The volume of a cylinder is given by $$ V = \pi r^2 h $$ and the surface area to be painted (the lateral surface area plus the top) is $$ A = 2\pi r h + \pi r^2 . $$Since the volume $$ V = 50  m$$³ is given, express the height $$ h  in $$terms of the radius $$ r $$ using the volume formula: $$ h = \frac{V}{\pi r^2} = \frac{50}{\pi r^2} . $$Substitute $$ h $$ from the previous step into the surface area formula to express $$ A $$ solely in terms of $$ r $$: $$ A(r) = 2\pi r \left(\frac{50}{\pi r^2}\right) + \pi r^2 . $$Simplify this expression. Differentiate the simplified surface area function $$ A(r) $$ with respect to $$ r  to $$find $$ A'(r) . $$This will help identify the critical points where the surface area could be minimized. Set the derivative $$ A'(r) $$ equal to zero and solve for $$ r  to $$find the critical points. Use the second derivative test or analyze the behavior of $$ A'(r)  to $$confirm that the critical point corresponds to a minimum surface area. Once $$ r  is $$found, use the expression for $$ h  to $$find the corresponding height.

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Concetti chiave

Volume of a Cylinder

Surface Area of a Cylinder

Optimization Techniques