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?

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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.
The price can decrease by up to approximately \$2,067 per month before revenue starts to drop.

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?

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