A length of ribbon 60 cm long is to be used to outline the boundary of a planar region
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
A length of ribbon 60 cm long is to be used to outline the boundary of a planar region, and every point on this boundary will serve as the base of a perpendicular square cross section in a three-dimensional solid. Specifically, you will form a rectangle of width w cm and length 60−2w cm (0<w<15), and for each x in the interval [0,w] the cross section perpendicular to the x–axis is a square whose side length equals the height of the rectangle at that x (that is, the constant length 60−2w). A solid S results.
(a) Write an expression for the volume V(w) of the solid S as a function of w. Clearly indicate the limits of integration.
(b) Determine the value of w that maximizes V(w). Show all analytic work, including verification that the value you find yields a maximum volume rather than a minimum.
(c) Compute the maximal volume of the solid, giving appropriate units.