Question

The sides of an equilateral triangle are increasing at a rate of 10 cm/min. at what rate is the area of the triangle increasing when the sides are 30 cm long?

Answer 1

Area of a triangle is given by

$Area= \frac{1}{2} ab\sin C= \frac{1}{2} s^2\sin C$

Since the sides of an equilateral triangle are equal.

Differentiating the area of the triangle, we have:

$\frac{dA}{dt} = \frac{dA}{ds} \cdot \frac{ds}{dt} =(s\sin C) \frac{ds}{dt}$

where

$\frac{ds}{dt} =10\,cm/min \\ s=30 \, cm$

$\therefore \frac{dA}{dt} =(30\sin60)\times10=300\sin60=259.8\, cm^2/min$

Therefore, the rate is the area of the triangle increasing when the sides are 30 cm long is 259.8 cm^2 / min