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Total revenue represents the total income generated from selling a certain number of units of a product. It plays a crucial role in understanding a business's financial health.

For our exercise, the total revenue function is given as:

• \( R(x) = 280x - 0.4x^2 \)

This formula signifies that total revenue depends on two factors: a linearly increasing part, seen in \( 280x \), and a diminishing return effect, represented by \( -0.4x^2 \).
To find how revenue changes with respect to time, we need to use calculus. Differentiating the revenue function with respect to \( x \) gives us the rate of change of revenue per unit sold:

• \( \frac{dR}{dx} = 280 - 0.8x \)

Next, applying the chain rule allows us to convert this rate into a time-dependent rate:

• \( \frac{dR}{dt} = (280 - 0.8x) \times \frac{dx}{dt} \)

When evaluated at \( x = 200 \) with \( \frac{dx}{dt} = 300 \), it results in:

• \( \frac{dR}{dt} = 12000 \) dollars per day

This indicates that at 200 units, the revenue increases by $12,000 every day as production increases by 300 units a day.

Total cost is the complete expense incurred in producing a set number of goods. Understanding cost trends is essential to maintaining operational efficiency.

For this case, the total cost function is:

• \( C(x) = 5000 + 0.6x^2 \)

This expression includes a base cost of \(5,000, plus a variable component growing quadratically with production, \( 0.6x^2 \).
Differentiating the cost function with respect to \( x \) reveals the cost change per additional unit:

• \( \frac{dC}{dx} = 1.2x \)

Utilizing the chain rule allows conversion to a time-dependent cost change:

• \( \frac{dC}{dt} = 1.2x \times \frac{dx}{dt} \)

When you substitute \( x = 200 \) and \( \frac{dx}{dt} = 300 \), this becomes:

• \( \frac{dC}{dt} = 72000 \) dollars per day

So, the cost increases by \)72,000 each day given a production rate increase of 300 units per day.

Profit represents the net earnings of a business after subtracting all costs from the revenue. Studying the rate of change of profit helps businesses anticipate financial trends.

In this context, profit is simply revenue minus cost:

• \( P(x) = R(x) - C(x) \)

The rate of profit change over time is therefore determined by:

• \( \frac{dP}{dt} = \frac{dR}{dt} - \frac{dC}{dt} \)

With previously calculated values for revenue and cost rates:

• \( \frac{dR}{dt} = 12000 \) dollars per day
• \( \frac{dC}{dt} = 72000 \) dollars per day

The result for the profit rate of change becomes:

• \( \frac{dP}{dt} = 12000 - 72000 = -60000 \) dollars per day

This negative value indicates a loss rate of $60,000 per day, suggesting a need to reassess production costs or pricing strategies to achieve profitability.