Question

A gardener is digging a plot of land. As he gets tired, he works more slowly. After $' t '$ 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,
$\frac{d A}{d t}=\frac{2}{\sqrt{t}}$
$\Rightarrow d A =\frac{2}{\sqrt{t}} d t$
On integrating both sides, we get
$\int d A=\int \frac{2}{\sqrt{t}} d t$
$A=\frac{2 t^{1 / 2}}{1 / 2}+C$
$\Rightarrow A=4 \sqrt{t}+C$
When $t=0, A=0$ then $C=0$
$\because A=4 \sqrt{t} \text { but } A=40$
$\therefore 40=4 \sqrt{t} \Rightarrow \sqrt{t}=10$
$\Rightarrow t=100\,min$