Questions: A manufacturer sells 40 boats per month at 45000 per boat, and each month demand is increasing at a rate of 3 boats per month. What is the fastest the price could drop before the monthly revenue starts to drop? [Revenue = Price × Quantity]

Solution Steps

To determine the fastest the price could drop before the monthly revenue starts to drop, we need to find the rate of change of revenue with respect to price and set it to zero. This involves calculating the derivative of the revenue function and solving for the price drop rate.

The revenue \( R \) is given by the product of price \( P \) and quantity \( Q \): \[ R = P \times Q \]

Given that the initial quantity is 40 boats and the demand is increasing at a rate of 3 boats per month, the quantity \( Q \) can be expressed as: \[ Q = 40 + 3 = 43 \]

To find the rate at which the price can drop before the revenue starts to drop, we need to differentiate the revenue function with respect to price \( P \): \[ \frac{dR}{dP} = Q \]

For the revenue to start dropping, the derivative of the revenue with respect to price must be zero: \[ \frac{dR}{dP} = 43 \]

Since the derivative is a constant (43), it implies that the revenue is always increasing with respect to price. Therefore, there is no rate at which the price can drop that will cause the revenue to start dropping.

Final Answer