Let's break this problem down to calculate the work done lifting both the bucket and the rope to the top of the 15-meter-tall building.

Given:

• Initial mass of the bucket (with sand): $m_b = 10 \, \text{kg}$
• Final mass of the bucket (with sand): $m_f = 6 \, \text{kg}$
• Mass of the rope per meter: $m_r = 0.3 \, \text{kg/m}$
• Height of the building: $h = 15 \, \text{m}$

Concept:

1. Variable Mass of the Bucket: The bucket loses mass as sand leaks at a constant rate, dropping from 10 kg to 6 kg. Thus, we can treat the mass as a function of height, with a linear decrease from $m_b$ to $m_f$.

2. Work Done on the Bucket: The total mass of the bucket at height $y$ is linearly decreasing with respect to $y$. Let $m(y)$ represent the mass at height $y$: $m(y) = m_b - \left(\frac{m_b - m_f}{h}\right) \cdot y = 10 - \left(\frac{10 - 6}{15}\right) \cdot y = 10 - \frac{4}{15}y$ The work done to lift the bucket is given by: $W_b = \int_0^h m(y) g \, dy$ where $g = 9.8 \, \text{m/s}^2$ is the acceleration due to gravity.

3. Work Done on the Rope: The rope has a constant mass per unit length, so the work done to lift a small section of rope to height $y$ is proportional to both the mass and height. The work to lift the entire rope is: $W_r = \int_0^h m_r y g \, dy$

Now, let’s calculate the total work done. I'll compute these integrals.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you still want to discuss this, feel free to ask me any details, and I can guide you through solving it step by step manually.

Would you like to:

1. Solve the integrals step-by-step manually?
2. Understand how work is calculated for each part (bucket and rope)?
3. Learn about variable mass problems in general?
4. Discuss the physical principles behind work and energy in lifting problems?
5. Explore other examples of mass-related work calculations?

Tip: When dealing with integrals, keep an eye on whether the mass or force is constant or variable over distance!