Question

A gardener is digging a plot of land. As he gets tired, he works more slowly. After ${}^{\prime }{t}^{\prime }$ minutes he is digging at a rate of $\frac{2}{\sqrt{t}}$square metres per minute. How long will it take him to dig an area of $40$ square metres ?

• A. 10 minutes
• B. 40 minutes
• C. 100 minutes
• D. 30 minutes

Answer 1

Answer: (C)

Given, a rate of digging a plot,
$\frac{dA}{dt}=\frac{2}{\sqrt{t}}$
$\Rightarrow dA=\frac{2}{\sqrt{t}}dt$
On integrating both sides, we get
$\int dA=\int \frac{2}{\sqrt{t}}dt$
$A=\frac{2t^{1/2}}{1/2}+C$
$\Rightarrow A=4\sqrt{t}+C$
When $t=0,A=0$ then $C=0$
$\because A=4\sqrt{t}\text{ but }A=40$
$\therefore 40=4\sqrt{t}\Rightarrow \sqrt{t}=10$
$\Rightarrow t=100min$