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A pen against a wall

CalculusEasy

A farmer has $84$ m of fencing to build a rectangular pen along a long straight wall, so only $3$ sides need fencing. What is the largest possible area of the pen, in m²? Give the answer as an integer.

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Hint

Take the side perpendicular to the wall as the variable and write the area as a quadratic.

Solution

Let each side perpendicular to the wall be $y$ ; the side parallel to it is then $84 - 2 y$ .
The area is $A (y) = y (84 - 2 y) = 84 y - 2 y^{2}$ .
Then $A^{'} (y) = 84 - 4 y$ , which is zero at $y = 21$ , and the parabola opens downward, so this is the maximum.
The pen is $21$ by $84 - 42 = 42$ , and its area is $21 \times 42 = 882$ .

Answer: $882$

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