# Algebra word problem

• Apr 6th 2008, 05:30 PM
gumi
Algebra word problem
A framer wants to enclose a rectangular field by a fence divide it into two smaller rectangular fields by constructing another fence parallel to one side of the field.
The farmer has 3000 yards of fencing. Find the dimension of the field so that total enclosed area is a maximum. (hint let h be the height and w be the width)

then 3h+2w=3000 You want to maximize the area hw. If you solve for h in terms of w then substitute into the expression hw, you get a quadratic function (you could just as well solve for w in terms of h) Find the maximum of quadratic using one of three techniques)
• Apr 6th 2008, 08:04 PM
gumi
bump
• Apr 6th 2008, 08:06 PM
Mathstud28
You could
either take the derivative then imput the values into $f''(x)$ and see if its negative values so you have maxes...test them in $f(x)$ and see what the absolute max is...or you can graph it and see the absolute max
• Apr 6th 2008, 08:56 PM
TheEmptySet
Quote:

Originally Posted by gumi
A framer wants to enclose a rectangular field by a fence divide it into two smaller rectangular fields by constructing another fence parallel to one side of the field.
The farmer has 3000 yards of fencing. Find the dimension of the field so that total enclosed area is a maximum. (hint let h be the height and w be the width)

then 3h+2w=3000 You want to maximize the area hw. If you solve for h in terms of w then substitute into the expression hw, you get a quadratic function (you could just as well solve for w in terms of h) Find the maximum of quadratic using one of three techniques)

Solving perimeter equation for h we get
$3h+2w=3000 \iff h=\frac{2}{3}(1500-w)$

subbing into the area equation

$A=hw=\frac{2}{3}(1500-w)w=-\frac{2}{3}w^2)+1000w$

So we note that this is a parabola and since they are symmetric about their vertex the max in half way inbetween the w intercepts. The w intercepts are w=0 and w=1500 so the vertex is at w=750.

we could also use the vertex formula

$w=\frac{-b}{2a}=\frac{-1000}{2\cdot \frac{-2}{3}}=\frac{-1000}{-\frac{4}{3}}=\frac{-1000}{1} \cdot \frac{-3}{4}=750$

You could also complete the square if you want.

Good luck.