Let's break this down step by step.

1. Find the rate of change of monthly sales: $q'(t)$

The monthly sales function is given by: $q(t) = 2000t - 100t^2$

To find the rate of change of sales, we differentiate $q(t)$ with respect to $t$:

$q'(t) = \frac{d}{dt}(2000t - 100t^2)$ $q'(t) = 2000 - 200t$

So, the rate of change of monthly sales is: $q'(t) = 2000 - 200t$

2. Find the rate of change of the price: $p'(t)$

The price function is given by: $p(t) = 1000 - t^2$

We differentiate $p(t)$ with respect to $t$:

$p'(t) = \frac{d}{dt}(1000 - t^2)$ $p'(t) = -2t$

So, the rate of change of the price is: $p'(t) = -2t$

3. Find the rate of change of monthly revenue 5 months after the introduction of the sound system

Revenue $R(t)$ is the product of price $p(t)$ and quantity sold $q(t)$:

$R(t) = p(t) \times q(t)$

Substitute the expressions for $p(t)$ and $q(t)$:

$R(t) = (1000 - t^2)(2000t - 100t^2)$

We now differentiate $R(t)$ with respect to $t$. First, let's simplify the expression for $R(t)$ by expanding the product:

$R(t) = (1000 \times 2000t) - (1000 \times 100t^2) - (t^2 \times 2000t) + (t^2 \times 100t^2)$ $R(t) = 2,000,000t - 100,000t^2 - 2000t^3 + 100t^4$

Now, differentiate $R(t)$ with respect to $t$:

$R'(t) = 2,000,000 - 200,000t - 6000t^2 + 400t^3$

Next, substitute $t = 5$ months into the derivative:

$R'(5) = 2,000,000 - 200,000(5) - 6000(5^2) + 400(5^3)$ $R'(5) = 2,000,000 - 1,000,000 - 6000(25) + 400(125)$ $R'(5) = 2,000,000 - 1,000,000 - 150,000 + 50,000$ $R'(5) = 900,000 \text{ dollars per month}$

4. Interpretation

When $t = 5$ months, the revenue is increasing at a rate of $900,000$ dollars per month.

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Would you like more details on any of these calculations or explanations? Here are some related questions for further practice:

1. What would the rate of change of sales be at $t = 10$ months?
2. How does the price change after 6 months? Calculate $p'(6)$.
3. Find the revenue at $t = 5$ months (without differentiating).
4. If the rate of change of price is negative, what does this mean in terms of business strategy?
5. How does the rate of change of revenue behave as $t$ increases?

Tip: Always check whether the price and sales functions reflect real-world behavior over time, especially if long-term predictions are needed.