A farmer wishes to construct a rectangular grazing pen for goats along a straight river

MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives

Problem

A farmer wishes to construct a rectangular grazing pen for goats along a straight river. The river forms one side of the rectangle, so fencing is required only for the remaining three sides. The farmer has exactly

200\,\text{m}

of fencing material available and wants the pen to have the greatest possible area. (a) Let

x

be the length of each side perpendicular to the river. Express the total area

A

of the pen as a function of

x

. Specify the domain of this function in the farming context. (b) Find all critical values of

A

on the domain found in part (a). Show all necessary steps in your calculation. (c) Determine the dimensions of the pen that maximize the area. Support your answer by analyzing the sign of

A'(x)

on appropriate intervals or by another rigorous argument. (d) Compute the maximum attainable area and state your answer with correct units. (e) The farmer considers setting aside

20\,\text{m}

of fencing for a gate, reducing the available fencing for the pen to

180\,\text{m}

. Without repeating all calculations, explain qualitatively how the optimal value of

x

from part (c) would change (increase, decrease, or remain the same). Justify your reasoning.

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