A rectangular lot is bordered on one side by a stream and on the other three sides by 600m of fencing. Find the dimensions of the lot if area is a maximum.
3 answers
150x300
May i ask how you found it?
you might look at the many similar questions below. I'll give you the shortcut, which always works in this kind of situation.
The maximum area is achieved when the fencing is divided equally among the lengths and the widths. In this case, we have one length and two widths. Half of 600 is 300, so there is one length of 300, and two widths of 150 each.
This also works where the area is divided into a grid of equal-size pens, each of width x and length y.
If there are m widths of x, and n lengths of y, then if the fencing has length z,
mx+ny = z
The area is mn(xy)
= mn(x)(z-mx)/n
= mx(z-mx)
= mzx - m^2 x^2
This is just a parabola, with vertex at x = mz/2m^2 = z/2m
That is, the m widths add up to z/2, or half the fencing.
The maximum area is achieved when the fencing is divided equally among the lengths and the widths. In this case, we have one length and two widths. Half of 600 is 300, so there is one length of 300, and two widths of 150 each.
This also works where the area is divided into a grid of equal-size pens, each of width x and length y.
If there are m widths of x, and n lengths of y, then if the fencing has length z,
mx+ny = z
The area is mn(xy)
= mn(x)(z-mx)/n
= mx(z-mx)
= mzx - m^2 x^2
This is just a parabola, with vertex at x = mz/2m^2 = z/2m
That is, the m widths add up to z/2, or half the fencing.
2 answers