A rectangular garden is to be constructed adjacent to a straight river
MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives
Problem
A rectangular garden is to be constructed adjacent to a straight river, so no fencing is needed along the side bordering the river. The side parallel to the river will use decorative fencing costing $30 per foot, while the two sides perpendicular to the river will use standard fencing costing $10 per foot. The owner has budgeted exactly $2400 for fencing material. Assume the side parallel to the river has length x feet and each side perpendicular to the river has length y feet.
(a) Express the total fencing cost C in terms of x and y, and write the budget condition that must be satisfied.
(b) Write the garden area A in terms of x and y. Use the budget condition from part (a) to rewrite A as a function of a single variable.
(c) Determine all critical points of the single-variable area function found in part (b), and state the domain over which the function is defined in the context of the problem.
(d) Show which critical point yields the maximum garden area, and find the corresponding dimensions x and y. Give numerical values to the nearest tenth of a foot.
(e) Compute the maximum possible area of the garden and interpret what the final result tells the owner about the relative lengths of the decorative and standard fences.