Question

The side of an equilateral triangle is increasing at the rate of $2cm/s$.At what rate is its area increasing ,when the side of the triangle is $20cm$

Answer 1

Area of equilateral triangle=$\frac{\sqrt{3}}{4}{a}^2$

where a=side of triangle

$⟹A=\frac{\sqrt{3}}{4}{a}^2$

differentiating with respect to time

$⟹\frac{dA}{dt}=\frac{\sqrt{3}}{4}\left(2a\right)\frac{da}{dt}$

$\because \frac{da}{dt}=2cm/secanda=20cm$

$⟹\frac{dA}{dt}=\frac{\sqrt{3}}{4}(2\times 30)\left(2\right)$

$=\frac{\sqrt{3}}{4}\times 40\times 2$

$=2\sqrt{3}\times 10$

$=20\sqrt{3}c{m}^2/sec$