Question

Each side of an equilateral triangle is increasing at the rate of 8 cm/hr. The rate of increase of its area when side is 2 cm, is

(a) $8\sqrt{3}{\mathrm{cm}}^2/\mathrm{hr}$

(b) $4\sqrt{3}{\mathrm{cm}}^2/\mathrm{hr}$

(c) $\frac{\sqrt{3}}{8}{\mathrm{cm}}^2/\mathrm{hr}$

(d) none of these

Answer 1

(a) $8\sqrt{3}{\mathrm{cm}}^2/\mathrm{hr}$

$$\begin{gathered} \text{Let}\mathit{\text{x}}\text{be the side and}\mathit{\text{A}}\text{be the area of the equilateral triangle at any time}\mathit{\text{t}}\text{. Then,} \\ A=\frac{\sqrt{3}}{4}{\mathit{\text{x}}}^2 \\ \Rightarrow \frac{dA}{dt}=\frac{\sqrt{3}}{2}x\left(\frac{dx}{dt}\right) \\ \Rightarrow \frac{dA}{dt}=\frac{\sqrt{3}}{2}\left(2\right)\left(8\right) \\ \Rightarrow \frac{dA}{dt}=8\sqrt{3}{\mathrm{cm}}^2/\mathrm{hr} \end{gathered}$$