A rectangular garden will be built against a straight river
MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives
Problem
A rectangular garden will be built against a straight river, so no fencing is required along the side bordering the water. The two sides perpendicular to the river use reinforced posts that cost 12 dollars per foot. The side opposite the river uses standard posts that cost 8 dollars per foot for the first 200 feet purchased; any length beyond 200 feet of standard fencing costs 5 dollars per foot. The garden must enclose 24,000 square feet of area.
(a) Let x be the length, in feet, of the sides perpendicular to the river. Express the total fence cost C(x) as a piecewise-defined function of x, indicating the domain for each piece.
(b) Compute C′(x) for every differentiable portion of C and determine all critical numbers that satisfy the domain restrictions.
(c) Identify the set of candidate values for x, evaluate C at each candidate, and determine the garden dimensions that minimize the total cost.
(d) Provide a justification that your dimensions indeed yield the minimum possible cost, making explicit reference to comparisons among all candidate values and any relevant endpoints.