A rectangular field is to be fenced on all four sides with exactly 40 meters of…

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

A rectangular field is to be fenced on all four sides with exactly 40 meters of fencing material. Let

x

meters denote the length of one side of the rectangle, and let

A(x)=x(20-x)

represent the area (in square meters) enclosed by the rectangle in terms of

x

. (a) Describe the practical meaning of

A'(x)

for this situation, including its units. (b) Explain why any interior maximum or minimum value of the area must occur at a value of

x

for which

A'(x)=0

. (c) Show that

A''\!(x)<0

at the critical point of

A(x)

and interpret this fact in the context of the field.