Question

A well in a circular area of radius $1000\mathrm{m}$ appears to lead to a lowering of the groundwater table (a drawdown) of $1\mathrm{m}$ at a distance of $10\mathrm{m}$ from the well. What is the drawdown at a distance of $100\mathrm{m}$ ?

Short answer

Answer: The drawdown at a distance of 100 meters from the well is 0.1 meters.

Step-by-step solution

Analyze the given data

We are given the following data: - Radius of the circular area: $1000\mathrm{m}$ - Drawdown at $10\mathrm{m}$: $1\mathrm{m}$ Our objective is to find the drawdown at a distance of $100\mathrm{m}$.

Assume a proportionality relationship

Let's assume that the drawdown ($D$) is inversely proportional to the distance ($r$) from the well. That is: $$D=k\frac{1}{r}$$ Where $k$ is the proportionality constant.

Determine the value of the proportionality constant

Using the given data, we can determine the proportionality constant. When the distance $r=10\mathrm{m}$, the drawdown $D=1\mathrm{m}$. So we get: $$1=k\frac{1}{10}$$ Solving for $k$ gives us: $$k=10$$

Find the drawdown at the required distance

Now, we want to find the drawdown at a distance of $100\mathrm{m}$. We do this by plugging the value of the proportionality constant in our earlier equation and let $r=100\mathrm{m}$: $$D=10\frac{1}{100}$$

Calculate the drawdown

Now, we can calculate the drawdown at a distance of $100\mathrm{m}$. $$D=\frac{10}{100}$$ $$D=0.1\mathrm{m}$$ So, the drawdown at a distance of $100\mathrm{m}$ is $0.1\mathrm{m}$.