Related Rates: Water Flowing into a Cylindrical Tank (Calculus)

Problem Setup: Water Flowing into a Cylinder

• 💡 The problem involves a cylindrical tank where water flows in at a specific rate.
• 🎯 We are given the rate of change of volume (dV/dt) as 40 cubic feet per minute.
• 🔑 We are also given the radius of the cylinder as a constant 6 ft.
• ❓ The goal is to find the rate at which the height of the water level is changing (dH/dt).

Volume Formula and Differentiation

• 📐 The formula for the volume of a cylinder is V = πr²h.
• ⚙️ Since the radius (r) is constant, we differentiate the volume formula with respect to time (t).
• 📈 The derivative of V with respect to t is dV/dt.
• 🚀 The derivative of πr²h with respect to t is πr²(dH/dt), as r is constant.

Solving for the Rate of Height Change

• 🧮 Rearranging the differentiated equation to solve for dH/dt, we get dH/dt = (dV/dt) / (πr²).
• 🧮 Substituting the given values: dH/dt = 40 / (π * 6²).
• 🧮 This simplifies to dH/dt = 40 / (36π), which further reduces to 10 / (9π) feet per minute.
• 📊 The height of the water level is increasing at a rate of approximately 0.354 feet per minute.