A manufacturer sells 40 boats per month at 22000 dollars per boat, and each month demand is increasing at a rate of 6 boats per month. What is the fastest the price col drop before the monthly revenue starts to drop?

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To determine the fastest rate at which the price of the boats can drop before the monthly revenue starts to decline, we'll analyze the relationship between price, quantity sold, and revenue.

### Given:
- **Initial Quantity Sold (q₀):** 40 boats/month
- **Initial Price (p₀):** $22,000 per boat
- **Rate of Increase in Demand (dq/dt):** 6 boats/month

### Revenue Formula:
$$ \text{Revenue (R)} = \text{Price (p)} \times \text{Quantity (q)} $$
$$ R = p \times q $$

### To Prevent Revenue Decrease:
We need to ensure that the rate of change of revenue (dR/dt) is **not negative**.

Differentiating the revenue with respect to time:
$$ \frac{dR}{dt} = \frac{dp}{dt} \times q + p \times \frac{dq}{dt} $$

To prevent revenue from decreasing:
$$ \frac{dR}{dt} \geq 0 $$
$$ \frac{dp}{dt} \times q + p \times \frac{dq}{dt} \geq 0 $$

Solving for $$\frac{dp}{dt}$$:
$$ \frac{dp}{dt} \geq - \frac{p \times \frac{dq}{dt}}{q} $$

### Plugging in the Numbers:
$$ \frac{dp}{dt} \geq - \frac{22,000 \times 6}{40} $$
$$ \frac{dp}{dt} \geq - \frac{132,000}{40} $$
$$ \frac{dp}{dt} \geq -3,300 \ \text{dollars/month} $$

### Conclusion:
The price can **decrease by up to $3,300 per month** without causing the monthly revenue to decline. If the price drops any faster than this, the revenue will start to decrease.

**Final Answer:**  
The price can fall by as much as $3 300 per month before revenue begins to decline.
The price can decrease by up to $3,300 per month before revenue starts to drop.

A manufacturer sells 40 boats per month at 22000 dollars per boat, and each month demand is increasing at a rate of 6 boats per month. What is the fastest the price col drop before the monthly revenue starts to drop?

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