Problem 46 - Calculus and its applications

Question

Great Green, Inc., determines that its marginal revenue per day is given by $$ R^{\prime}(t)=75 e^{t}-2 t, \quad R(0)=0 $$ where \(R(t)\) is the total accumulated revenue, in dollars, on the tth day. The company's marginal cost per day is given by $$ C^{\prime}(t)=75-3 t, \quad C(0)=0 $$ where \(C(t)\) is the total accumulated cost, in dollars, on the \(t\) th day. a) Find the total profit from \(t=0\) to \(t=10\) (see Exercise 45). b) Find the average daily profit for the first 10 days.

Step-by-Step Solution

Answer

Total profit for 10 days is calculated using \(P(10) - P(0)\). Average profit is this total divided by 10.

Step 1: Understand the Context

Great Green, Inc. has a marginal revenue function \(R'(t)\) and a marginal cost function \(C'(t)\). We need to find the total profit over the first 10 days and then determine the average daily profit for these 10 days.

Step 2: Integrate the Marginal Revenue

The total revenue \(R(t)\) can be calculated by integrating the marginal revenue function \(R'(t) = 75e^t - 2t\). \[R(t) = \int (75e^t - 2t) \, dt\]Calculating the integral:\(R(t) = 75e^t - t^2 + C_1\). Given \(R(0) = 0\): \[75e^0 - 0^2 + C_1 = 0\] Thus, \(C_1 = -75\). So, \(R(t) = 75e^t - t^2 - 75\).

Step 3: Integrate the Marginal Cost

Similarly, the total cost \(C(t)\) can be calculated by integrating the marginal cost function \(C'(t) = 75 - 3t\).\[C(t) = \int (75 - 3t) \, dt\]Calculating the integral:\(C(t) = 75t - \frac{3t^2}{2} + C_2\). Given \(C(0) = 0\):\[75 \times 0 - \frac{3 \times 0^2}{2} + C_2 = 0\]Thus, \(C_2 = 0\). So \(C(t) = 75t - \frac{3t^2}{2}\).

Step 4: Calculate Total Profit from t=0 to t=10

Total profit \(P(t)\) is the difference between total revenue and total cost: \[P(t) = R(t) - C(t)\]Thus:\[P(t) = \left(75e^t - t^2 - 75\right) - \left(75t - \frac{3t^2}{2}\right)\]Simplifying, \[P(t) = 75e^t - \frac{1}{2}t^2 - 75t - 75\]Calculate \(P(10) - P(0)\):For \(t=10\): \[P(10) = 75e^{10} - \frac{1}{2}(10^2) - 75(10) - 75\]Evaluating:\[P(10) = 75e^{10} - 50 - 750 - 75\]For \(t=0\):\[P(0) = 75e^0 - \frac{1}{2}(0^2) - 75(0) - 75 = 0 - 75\]Thus:\(P(10) - P(0)\) gives total profit over 10 days.

Step 5: Calculate Average Daily Profit for First 10 Days

The average daily profit over the first 10 days is: \[\text{Average Profit} = \frac{\text{Total Profit over 10 days}}{10}\]This gives us the mean profit per day for the period.

Key Concepts

Understanding Marginal Revenue

Marginal revenue reflects how much an additional unit of product adds to total revenue. In mathematical terms, this is the derivative of the revenue function, denoted as \( R'(t) \). In the context of Great Green, Inc., the marginal revenue equation \( R'(t) = 75e^t - 2t \) helps the company understand how each additional day affects their revenue stream.

Calculating the total revenue requires integrating the marginal revenue function over the desired time period. By performing this integration, we essentially "add up" all the small revenue changes from day to day, giving us the total revenue function \( R(t) \) over time.

• The importance of marginal revenue lies in decision making, helping firms optimize their production levels.
• By adjusting output until marginal revenue equals marginal cost, companies can maximize profit.

Exploring Marginal Cost

Marginal cost is the cost added by producing one more unit of a good. For Great Green, Inc., it's represented as \( C'(t) = 75 - 3t \), indicating how costs evolve daily. Understanding this concept is crucial for firms to manage expenses and profit margins effectively.

Integration of the marginal cost function allows us to find the total cost \( C(t) \). It shows the sum of all small cost increments over time. Obtaining \( C(t) \) from \( C'(t) \) helps in detailing costs accumulated by certain business activities up to specific time points.

• Marginal cost informs production decisions—too high a marginal cost can indicate diminishing returns.
• Firms constantly compare their marginal cost against marginal revenue to determine optimal pricing and output levels.

The Role of Integration in Calculus Applications

Integration is a critical tool in calculus used to find the total or accumulated value from a rate of change over time, such as revenue and cost functions. When dealing with functions like \( R'(t) \) and \( C'(t) \), integration is necessary to obtain the complete revenue \( R(t) \) and cost \( C(t) \) functions respectively.

In business applications:

• Integration helps sum daily changes to find a cumulative total.
• It's essential for computing comprehensive financial metrics like total revenue, cost, and profit over time.
• This mathematical operation transforms marginal metrics into total metrics.

These integrations provide vital insights into cumulative outcomes, such as the total profit over a period, highlighting the power of calculus in real-world applications.