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 price decrease that keeps the revenue constant or increasing. Revenue is calculated as the product of price and quantity. We will set up an equation for revenue and find the derivative with respect to price to determine the rate of price change that maintains or increases revenue.

The revenue \( R \) is given by the product of price \( P \) and quantity \( Q \). Initially, the price is \$45,000 and the quantity is 40 boats. The quantity increases by 3 boats per month. Therefore, the revenue function is: \[ R = P \times (40 + 3) = 43P \]

To find the rate of change of revenue with respect to price, we differentiate the revenue function with respect to \( P \): \[ \frac{dR}{dP} = 43 \]

The derivative \(\frac{dR}{dP} = 43\) is positive, indicating that the revenue increases as the price increases. Therefore, any decrease in price will result in a decrease in revenue.

Final Answer