Maximising Rainwater Harvesting

📌 CONCEPT: The problem involves finding the depth at which the rate of increase of the volume of water in a cylindrical tank is equal to the rate of increase of its surface area, using related rates.

📐 RULE / FORMULA: We will use the formulas V = πr²h and A = 2πr² + 2πrh, where V is the volume, A is the surface area, r is the radius, and h is the height.

💡 WORKED EXAMPLE: Let's consider a tank with a radius of 4 meters and a height of 6 meters. We need to find the depth at which dV/dt = dA/dt. Using the given formulas, we can differentiate both V and A with respect to time t. Then, we can set up the equation d(πr²h)/dt = d(2πr² + 2πrh)/dt and solve for h.

⚠️ COMMON MISTAKE: Students may incorrectly assume that the rates of change are equal at the same depth, without considering the time derivative and related rates.