Question

The sides of an equilateral triangle are decreasing at a rate of 3 in/hr. How fast is the area enclosed by the triangle decreasing when the sides are 2 feet long?

Answer 1

1 foot =12 inches, thus 2 feet =24 inches.

An equilateral triangle has all its side equal i.e s.

h= $\frac{✓3} {2}$s

A=$\frac {1}{2}×s×\frac{✓3} {2}$

=$\frac{1} {4}✓3$s²

If $\frac{ds} {dt}$ =-3in/hr,when s=24, $\frac{dA}{dt}$ =?

Differentiating with respect to t;

$\frac{dA} {dt}$ $= \frac{✓3} {4}$×2s $\frac{ds} {dt}$

Hence s=24 when $\frac{dA} {dt}$ =$\frac{✓3} {2}$ (24)(-3)=-36✓3 in/hr

=-36✓3 in/hr