Calculus I · Antiderivatives, definite integrals, and the Fundamental Theorem of Calculus
A farmer has 240 m of fencing and will enclose a rectangular pasture along a straight…
Problem
A farmer has \(240\,\mathrm{m}\) of fencing and will enclose a rectangular pasture along a straight river that needs no fence. Let \(x>0\) be the length, in meters, of each of the two sides perpendicular to the river, and let the side parallel to the river have length \(y>0\). (a) Use the fencing constraint to express the enclosed area as a function \(A(x)\) on a closed interval of physically possible \(x\), including the degenerate cases of zero area. (b) Find every critical point of \(A\) in the interior of that interval. (c) Determine the dimensions that maximize the enclosed area and the maximum area. Give exact values with units, and justify the maximum by comparing the interior critical value with the endpoint values.
Hint
Only three sides are fenced: two of length \(x\) and one of length \(y\) along the river.
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