# What is the rate of change of the width (in ft/sec) when the height is 10 feet, if the height is decreasing at that moment at the rate of 1 ft/sec.A rectangle has both a changing height and a changing width, but the height and width change so that the area of the rectangle is always 60 square feet?

Mar 16, 2015

The rate of change of the width with time $\frac{\mathrm{dW}}{\mathrm{dt}}$ = $0.6 \text{ft/s}$

$\frac{\mathrm{dW}}{\mathrm{dt}} = \frac{\mathrm{dW}}{\mathrm{dh}} \times \frac{\mathrm{dh}}{\mathrm{dt}}$

$\frac{\mathrm{dh}}{\mathrm{dt}} = - 1 \text{ft/s}$

So $\frac{\mathrm{dW}}{\mathrm{dt}} = \frac{\mathrm{dW}}{\mathrm{dh}} \times - 1 = - \frac{\mathrm{dW}}{\mathrm{dh}}$

$W \times h = 60$

$W = \frac{60}{h}$

$\frac{\mathrm{dW}}{\mathrm{dh}} = - \frac{60}{{h}^{2}}$

So $\frac{\mathrm{dW}}{\mathrm{dt}} = - \left(- \frac{60}{{h}^{2}}\right) = \frac{60}{{h}^{2}}$

So when $h = 10$:

$\Rightarrow$ $\frac{\mathrm{dW}}{\mathrm{dt}} = \frac{60}{{10}^{2}} = 0.6 \text{ft/s}$