# Maxima and minima problem

• Apr 9th 2012, 07:50 PM
julie9300
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
• Apr 9th 2012, 08:06 PM
Prove It
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 2km2 but I keep getting 1km and 1km

If we call \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 \begin{align*} 4 - 2x \end{align*}.

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

Now how would you find the maximum?
• Apr 9th 2012, 08:15 PM
julie9300
Re: Maxima and minima problem
Thanks! :)

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