Expert: Pawan Musale - 3/27/2009

Question
A rectangular lot is bounded on one side by a river and on the other three sides by a total of 80 m of fencing. Find the dimensions of the largest possible lot.

I've never encountered an equation like this. Where do I start?

Answer
Faisal,

We have to find the maximum possible area of the lot. Now say that one side of the lot, which is parallel to river side, is 'x'. The other two sides are say 'y'. See the attached drawing. From the explanation there, we get x = 80 - 2y.

So now we have got two sides which are y and 80 -2y. ------ (1)

Now remember that while solving these types of problems, always bring down two sides in one variable only (here 'y').

Now area (A) of the lot becomes,

A = y (80 - 2y)
A = 80y - 2y^2

Now this equation is polynomial of degree 2 (since power of y is 2).

REMEMBER: To find out maximum or minimum value of a polynomial equation, take derivative of that equation and equate it with 0.

So it becomes,
80 - 4y = 0
80 = 4y
y = 20      This is one side of plot --------------------- (2)

From statement (1) above, second side is 80 - 2y which comes out to be 40 ------------- (3)

Hence

dimensions of largest possible plot are 20 and 40.

I hope this is clear.

If you are not aware of derivatives, then please let me know. I can try to explain this concept.

All the best

Regards,
Pawan