A 384 - square meter plot of land is to be enclosed by a fence and divide into two equal parts by another fence parallel to one pair of sides. What dimensions of the outer rectangle will minimize the amount of fence used?
Take the sides to which the inner fence is parallel to as having length x.
Then the other two sides have length y.
Fence length = 3x+2y
Area =xy=384
Substitute x or y into the linear equation
$\displaystyle xy=384\ \Rightarrow\ x=\frac{384}{y}$
$\displaystyle \frac{3(384)}{y}+2y=k$
$\displaystyle 1152+2y^2=ky$
Differentiate and equate to zero to find minimum y
$\displaystyle 4y=k$
$\displaystyle 1152=2y^2$
$\displaystyle y=\sqrt{576}=24$
$\displaystyle x=\frac{384}{24}=16$
Hi deltax,
sorry, that was a few steps in one go!
$\displaystyle \frac{3(384)}{y}+2y=k$
$\displaystyle \frac{1152}{y}+2y=k$
multiplying by y
$\displaystyle 1152+2y^2=ky$ (1)
Differentiate
$\displaystyle 4y=k$
Now, use this instead of k in (1)
$\displaystyle 1152+2y^2=4y(y)=4y^2$
Subtract $\displaystyle 2y^2$ from both sides
$\displaystyle 1152+2y^2-2y^2=4y^2-2y^2$
$\displaystyle 1152=2y^2$
$\displaystyle y^2=\frac{1152}{2}=576$
$\displaystyle y=\sqrt{576}$