A landscape architect plans to build an open rectangular flower garden adjacent to a…
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A landscape architect plans to build an open rectangular flower garden adjacent to a straight brick wall that will serve as one side of the rectangle. The other three sides of the garden require wooden fencing. In addition, to divide the garden into two equal smaller rectangles, a divider fence parallel to the wall will be installed halfway across the width, as shown in the verbal description below.
• Let x meters denote the width perpendicular to the wall.
• Let y meters denote the length along the wall.
• Exactly 60 meters of wooden fencing are available for the three outer sides and the single divider.
(a) Express the total area A of the entire garden as a function of x only. State the domain of this function in the given context.
(b) Determine all critical values of A(x) within the domain found in part (a).
(c) Using an appropriate derivative test, justify that the critical value found in part (b) yields the maximum possible area of the garden.
(d) Compute the dimensions of the garden that maximize its area, and state the maximum area to the nearest tenth of a square meter.