Riverbank Erosion Rate?

📌 CONCEPT: The problem describes a situation where the rate of riverbank erosion is proportional to the square root of the length of the bank that has already eroded. This can be modeled using the concept of related rates and differential equations.

📐 RULE / FORMULA: Let's denote the length of the eroded bank as 'x' and the rate of erosion as 'dx/dt'. The given condition can be expressed as dx/dt = k√x, where 'k' is the constant of proportionality.

💡 WORKED EXAMPLE: Given that the bank erodes at a rate of 0.5 meters per day when half of it has eroded, we can substitute x = 50 and dx/dt = 0.5 into the equation to find the value of 'k'. After finding 'k', we can use it to calculate the rate of erosion after 10 days.

⚠️ COMMON MISTAKE: Students often forget to check the units of the given values and the derived value of 'k', which can lead to incorrect calculations and units.