AP Calculus AB · Analytical applications of differentiation
A closed right circular cylinder must enclose a volume of 250π cubic centimeters
Problem
A closed right circular cylinder must enclose a volume of \(250\pi\) cubic centimeters. Let \(r>0\) be the radius of the base, in centimeters, and let \(h>0\) be the height. The total surface area \(S\), including both circular ends, is to be minimized. (a) Use the volume constraint to express \(S\) as a function of \(r\) alone and state the domain \(r>0\). (b) Find the critical point of \(S\) in this domain. (c) Determine the radius and height that minimize \(S\), together with the minimum surface area. Justify that the critical point is a global minimum on \(r>0\) by analyzing the sign of \(S'\) or the limiting behavior of \(S\) as \(r\to 0^+\) and as \(r\to\infty\).
Hint
Eliminate the height with the fixed-volume constraint before differentiating the surface area.
Check your work
Work the problem yourself first. Then open it in Training to check your answer and read the full worked solution.
The answer check and full solution for this problem come with ProofAnvil Practice membership ($19 USD monthly). See membership. Or start with the free AP Calculus AB sample problem: Try the free sample problem.
More AP Calculus AB practice problems
- On the interval [-4,6], the graph of a function r consists of three line segments: the…Limits and continuity
- A continuous function T gives the outdoor temperature in degrees Fahrenheit t hours…Limits and continuity
- The curve x^2+3xy+2y^2=12 contains the point (2,1)Differentiation definitions and fundamental rules
- A drone flies horizontally at a constant altitude of 80 metersDifferentiation definitions and fundamental rules
- For 0<x<π, let y=(sin x)^xComposite, implicit, and inverse-function differentiation
- The cost, in dollars, of producing q liters of a chemical is given by a differentiable…Contextual applications of differentiation
- A well-stirred tank initially contains 80 liters of pure waterIntegration and accumulation of change
- A tank contains 100 gallons of water at time t=0 minutesDifferential equations
- Let R be the region bounded by y=4-x^2 and y=x+2Applications of integration