Problem 45 - Calculus and its applications

Question

Shylls, Inc., determines that its marginal revenue per day is given by $$ R^{\prime}(t)=100 e^{t}, \quad R(0)=0 $$ where \(R(t)\) is the total accumulated revenue, in dollars, on the tth day. The company's marginal cost per day is given by $$ C^{\prime}(t)=100-0.2 t, \quad C(0)=0 $$ where \(C(t)\) is the total accumulated cost, in dollars, on the \(t\) th day. a) Find the total profit from \(t=0\) to \(t=10\) (the first lo days). Note: $$ P(T)=R(T)-C(T)=\int_{0}^{T}\left[R^{\prime}(t)-C^{\prime}(t)\right] d t $$ b) Find the average daily profit for the first 10 days \((\) from \(t=0\) to \(t=10)\).

Step-by-Step Solution

Answer

The total profit for the first 10 days is approximately 22031 dollars. The average daily profit is 2203.1 dollars.

Step 1: Define Profit Function

Define the profit function as the difference between the total revenue and total cost. According to the problem, this is given by the integral:\[ P(T) = \int_{0}^{T} \left[ R'(t) - C'(t) \right] dt \]

Step 2: Substitute Revenue and Cost Derivatives

Substitute \( R'(t) = 100e^t \) and \( C'(t) = 100 - 0.2t \) into the profit function:\[ P(T) = \int_{0}^{T} \left[ 100e^t - (100 - 0.2t) \right] dt \]which simplifies to:\[ P(T) = \int_{0}^{T} \left[ 100e^t - 100 + 0.2t \right] dt \]

Step 3: Calculate the Indefinite Integral

Integrate term-by-term:1. \( \int 100e^t dt = 100e^t + C_1 \)2. \( \int (-100) dt = -100t + C_2 \)3. \( \int 0.2t dt = 0.1t^2 + C_3 \)Combine these results:\[ P(t) = [100e^t - 100t + 0.1t^2 + C] \]

Step 4: Evaluate the Definite Integral from t=0 to t=10

Compute \( P(10) - P(0) \):- \( P(10) = 100e^{10} - 100(10) + 0.1(10)^2 + C \)- \( P(0) = 100e^0 - 100(0) + 0.1(0)^2 + C = 100 \)Thus, \[ P(10) - P(0) = (100e^{10} - 1000 + 10) - 100 \]

Step 5: Simplify the Result

Compute the simplified expression:\[ P(10) - P(0) = 100e^{10} - 1090 \]The total profit from \( t=0 \) to \( t=10 \) is approximately \( 22031 \) dollars when substituting \( e \approx 2.718 \).

Step 6: Calculate Average Daily Profit

The average daily profit is given by dividing the total profit by the number of days:\[ \text{Average Daily Profit} = \frac{P(10) - P(0)}{10} \]Substituting the total profit:\[ \text{Average Daily Profit} = \frac{100e^{10} - 1090}{10} \approx 2203.1 \]

Key Concepts

Marginal Revenue

Marginal revenue represents the additional income generated from selling one more unit of a product or service. It plays a crucial role in deciding how much a company should produce to maximize its profits. In the exercise, the marginal revenue, denoted as \( R'(t) \), is given by the function \( 100e^t \). This indicates that the marginal revenue grows exponentially over time with the passage of each day.
The exponential function \( e^t \) suggests that marginal revenue increases rapidly as time progresses, reflecting a situation where each additional sale contributes significantly more to the revenue. Understanding this trend is essential for businesses to make informed production and pricing decisions.

• The initial condition \( R(0) = 0 \) shows that no revenue has been accumulated at the start (\( t = 0 \)).
• Businesses aim to operate where marginal revenue meets or exceeds marginal cost, ensuring profitability.

Marginal Cost

Marginal cost is the additional cost incurred in producing one more unit of a product. It is crucial for understanding the cost dynamics as production increases. In the exercise, the marginal cost, denoted as \( C'(t) \), is described by the function \( 100 - 0.2t \). This means over each day, the marginal cost decreases slightly, as expressed by the \(-0.2t\) term.
This declining nature suggests that, initially, producing more units incurs higher costs, but over time, each subsequent unit becomes slightly less expensive to produce. This typically occurs due to efficiencies gained through larger scale production.

• The initial condition \( C(0) = 0 \) suggests no production cost has been accumulated by day zero.
• Marginal cost is compared with marginal revenue to determine optimal production levels.

Definite Integral

The definite integral provides a method of computing the accumulation of quantities, which in this scenario is the profit over a set period. It represents the total accumulation of revenue minus cost across the days from \( t=0 \) to \( t=10 \). The definite integral \[ P(T) = \int_{0}^{T} \left[ R'(t) - C'(t) \right] dt \] captures this total profit.
By evaluating this integral from specific endpoints, namely \( t=0 \) to \( t=10 \), one can calculate the total profit for the first ten days. This profit is essentially the net revenue after accounting for costs over the given time interval.

• The integral captures the cumulative effect of both marginal revenue and marginal cost, taking into account their behaviors over time.
• The process of finding the definite integral involves calculating the accumulation of changes in revenue and cost, providing a comprehensive view of business performance over time.

Using calculus in this way helps businesses make crucial long-term decisions by highlighting trends that aren't immediately obvious through basic arithmetic calculations.