Question

A gardener is digging a plot of land.

As he gets tried, he works more slowly.

After $t$ min, he is digging at a rate of $\frac{2}{\sqrt{t}}$ m²/min.

How long will take him to dig an area of $40$ sqm?

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

Answer 1

The correct option is A

$100$minutes

The explanation for the correct answer:

Step 1: Finding the rate of digging the plot of land:

Area of digging = $40$sqm

Rate of digging after $t$ minutes = $\frac{2}{\sqrt{t}}$

$$\begin{gathered} \Rightarrow \frac{dA}{dt}=\frac{2}{\sqrt{t}} \\ \Rightarrow dA=\frac{2}{\sqrt{t}}dt \end{gathered}$$

Integrate both sides with respect to $t,$

$$\begin{gathered} \int dA=\int \frac{2}{\sqrt{t}}dt \\ A=\frac{\left(2\right){\left(t\right)}^{{\displaystyle \frac{1}{2}}}}{{\displaystyle \frac{1}{2}}}+C \\ A=4\sqrt{t}+C \end{gathered}$$

Step 2: Finding the time for digging:

When $t=0,A=0$then $C=0$.

But we know that $A=40$

Thus the equation becomes,

$$\begin{gathered} \Rightarrow 40=4\sqrt{t} \\ \Rightarrow \sqrt{t}=10 \\ \Rightarrow t={10}^2 \end{gathered}$$

$\Rightarrow t=100$minutes

Hence, option (A) is the correct option.