A manufacturer sells 75 boats per month at 31000 dollars per boat, and each month demand is increasing at a rate of 5 boats per month. What is the fastest the price could drop before the monthly revenue starts to drop?

To determine the fastest rate at which the price of boats can drop before the monthly revenue starts to decline, we'll analyze the relationship between price, quantity sold, and revenue. ### Given: - **Current Price (P):** \$31,000 per boat - **Current Quantity Sold (Q):** 75 boats per month - **Rate of Change of Quantity (\( \frac{dQ}{dt} \)):** +5 boats per month ### Revenue Formula: \[ \text{Revenue} (R) = P \times Q \] ### Rate of Change of Revenue: To ensure that revenue doesn't decline, the rate of change of revenue (\( \frac{dR}{dt} \)) should be non-negative. \[ \frac{dR}{dt} = P \frac{dQ}{dt} + Q \frac{dP}{dt} \geq 0 \] ### Solving for \( \frac{dP}{dt} \): \[ P \frac{dQ}{dt} + Q \frac{dP}{dt} \geq 0 \] \[ Q \frac{dP}{dt} \geq -P \frac{dQ}{dt} \] \[ \frac{dP}{dt} \geq -\frac{P}{Q} \frac{dQ}{dt} \] ### Plugging in the Numbers: \[ P = \$31,000 \] \[ Q = 75 \text{ boats} \] \[ \frac{dQ}{dt} = 5 \text{ boats/month} \] \[ \frac{dP}{dt} \geq -\frac{31,000}{75} \times 5 \] \[ \frac{dP}{dt} \geq -\frac{155,000}{75} \] \[ \frac{dP}{dt} \geq -2,066.67 \text{ dollars/month} \] ### Conclusion: The manufacturer can **decrease the price by up to approximately \$2,067 per month**. Any faster reduction in price would result in a decrease in monthly revenue. **Final Answer:** The price may fall by up to approximately $ 2 067 per month before revenue declines.