MATHEMATICS · Single correct · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
Which of the following best explains why, when maximizing the volume of a closed cylindrical can that must use exactly 500 cm2 of material, it is necessary to express the surface–area condition as an equation relating h and r before differentiating the volume?
(a) Using the relation makes the derivative test unnecessary because the maximum will always occur when h=r.
(b) The relation guarantees that the volume and surface area are simultaneously maximized at the same values of h and r.
(c) The relation reduces the number of independent variables so that a single-variable function can be analyzed for extrema.
(d) Without the relation, the derivative of the volume with respect to r would always be zero, giving no information about maxima.