09 - 11 Rectangular Lot Problems in Maxima and Minima

$0 = xy' + y$

$y' = -y/x$
 

Perimeter:
$P = 2x + 2y$

$dP/dx = 2 + 2y' = 0$

$1 + (-y/x) = 0$

$y = x$   (a square)           answer

$0 = xy' + y$

$y' = -y/x$
 

Perimeter:
$P = x + 2y$

$dP/dx = 1 + 2y' = 0$

$1 + 2(-y/x) = 0$

$y = \frac{1}{2}x$
 

width = ½ × length           answer

$k = mx + 2ny + nx$

$k - (m + n)x = 2ny$

$y = \dfrac{k}{2n} - \dfrac{m + n}{2n}x$
 

Area:
$A =xy$

$A = x\left( \dfrac{k}{2n} - \dfrac{m + n}{2n}x \right)$

$A = \dfrac{k}{2n}x - \dfrac{m + n}{2n}x^2$

$\dfrac{dA}{dx} = \dfrac{k}{2n} - \dfrac{m + n}{n}x = 0$

$\dfrac{k}{2n} = \dfrac{m + n}{n}x$

$x = \dfrac{k}{2(m + n)}$
 

$y = \dfrac{k}{2n} - \dfrac{m + n}{2n} \, \left[ \dfrac{k}{2(m + n)} \right]$

$y = \dfrac{k}{4n}$
 

$A = \dfrac{k}{2(m + n)} \times \dfrac{k}{4n}$

$A = \dfrac{k^2}{8n\,(m + n)} \, \text{ yard}^2$           answer