Maxima and minima problem

A farmer has 4 km of fencing wire and wishes to fence a rectangular piece of land through which flows a straight river, which is to be utilised as one side of the enclosure. How can this be done to enclose as much land as possible?

The actual answer is: __2 km x 1km = maximum of 2km__^{2} but I keep getting 1km and 1km

Re: Maxima and minima problem

Quote:

Originally Posted by

**julie9300** A farmer has 4 km of fencing wire and wishes to fence a rectangular piece of land through which flows a straight river, which is to be utilised as one side of the enclosure. How can this be done to enclose as much land as possible?

The actual answer is: __2 km x 1km = maximum of 2km__^{2} but I keep getting 1km and 1km

If we call $\displaystyle \displaystyle \begin{align*} x \end{align*}$ the width of the rectangle, and we know that the three sides of the rectangle that come from the fencing wire add up to 4km, then the length is $\displaystyle \displaystyle \begin{align*} 4 - 2x \end{align*}$.

Thus, the area of the land is $\displaystyle \displaystyle \begin{align*} A = x(4 - 2x) = 4x - 2x^2 \end{align*}$.

Now how would you find the maximum?

Re: Maxima and minima problem

Thanks! :)

I realised my mistake, I kept calculating using four sides rather than three