# Thread: Maxima and minima problem

1. ## 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 2km2 but I keep getting 1km and 1km

2. ## Re: Maxima and minima problem

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 2km2 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?

3. ## Re: Maxima and minima problem

Thanks!

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