Question
A wire of length 20 meters is used to form two sides of a rectangular enclosure, while the other two sides are formed by a river. What are the dimensions of the enclosure that maximizes its area?
A wire of length 20 meters is used to form two sides of a rectangular enclosure, while the other two sides are formed by a river. What are the dimensions of the enclosure that maximizes its area?
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Koala Tutor
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Let the width of the enclosure be $w$ and the length be $l$, so the area of the enclosure is $lw$. We need to maximize this area with the constraint $2w+l=20$. Solving the constraint for $l$, we get $l=20-2w$. Substituting this into the equation for the area, we have $A=w(20-2w)=20w-2w^2$. Completing the square, we have $A=-(2w-5)^2+25$. Since the maximum value of $-(2w-5)^2$ is 0 (when $w=\frac{5}{2}$), the maximum value of $A$ is $\boxed{25}$ square meters, achieved when $w=\frac{5}{2}$ and $l=20-2w=10$.
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