Let's work through this step by step:

Part (a) Determine the force applied to lift the bucket.

Initially, the total mass of the bucket with sand is 40 kg. As the bucket is lifted, sand leaks at a rate of 0.05 kg per meter. If the bucket is lifted by a distance $x$, the mass of the sand decreases linearly as:

$m(x) = 40 - 0.05x$

The force applied to lift the bucket at a height $x$ is given by the weight of the bucket and the remaining sand at that height. This force is:

$F(x) = m(x) \cdot g = (40 - 0.05x) \cdot 9.8$

Where $g = 9.8 \, \text{m/s}^2$ is the acceleration due to gravity. Thus:

$F(x) = (40 - 0.05x) \cdot 9.8 \, \text{N}$

Part (b) Work done to lift the bucket.

The work done, $W$, is the integral of the force over the distance lifted, which is from 0 to 20 meters:

$W = \int_0^{20} F(x) \, dx = \int_0^{20} (40 - 0.05x) \cdot 9.8 \, dx$

Let's compute this:

$W = 9.8 \int_0^{20} (40 - 0.05x) \, dx$

$W = 9.8 \left[ 40x - 0.05 \cdot \frac{x^2}{2} \right]_0^{20}$

Evaluating the integral:

$W = 9.8 \left[ (40 \cdot 20) - 0.05 \cdot \frac{20^2}{2} \right]$

$W = 9.8 \left[ 800 - 0.05 \cdot 200 \right]$

$W = 9.8 \left[ 800 - 10 \right] = 9.8 \times 790 = 7742 \, \text{Joules}$

So, the work done is approximately:

$W \approx 7742 \, \text{J}$

Do you have any questions or want further details?

Here are 5 related questions to expand your understanding:

1. What happens to the force if the sand leakage rate changes?
2. How would the work change if the height of the building were doubled?
3. What is the physical interpretation of the integral used in calculating the work done?
4. How would the problem change if the rope had a significant weight?
5. How would varying gravitational acceleration affect the force and work calculations?

Tip: When dealing with variable forces, using integrals to sum up the total effect is crucial for accurate work calculations.