Problem 35 - Applied Calculus

Question

Revenues at 3D Systems are predicted to be \(14.2 x^{2}+120 x+474\) million dollars per year, where \(x\) is the number of years since \(2013 .\) Predict the average annual revenue from 2013 to $2023 .

Step-by-Step Solution

Answer

The average annual revenue is approximately 1547.33 million dollars.

Step 1: Identifying the Problem

To find the average annual revenue from 2013 to 2023, we need to compute the average value of the function over the given interval. The function given is \( f(x) = 14.2x^2 + 120x + 474 \), and we need the average from \( x = 0 \) to \( x = 10 \) as \( x = 0 \) corresponds to 2013, and \( x = 10 \) corresponds to 2023.

Step 2: Setting Up the Average Value Formula

The formula to find the average value of a function \( f(x) \) over an interval \([a, b]\) is given by: \[ \text{Average} = \frac{1}{b-a} \int_{a}^{b} f(x) \, dx \]. In this scenario, \( a = 0 \) and \( b = 10 \), so we need to find \( \frac{1}{10 - 0} \int_{0}^{10} (14.2x^2 + 120x + 474) \, dx \).

Step 3: Calculating the Integral

We find \( \int (14.2x^2 + 120x + 474) \, dx \). The integral of \( 14.2x^2 \) is \( \frac{14.2}{3}x^3 \), the integral of \( 120x \) is \( 60x^2 \), and the integral of \( 474 \) is \( 474x \). So, \( \int (14.2x^2 + 120x + 474) \, dx = \frac{14.2}{3}x^3 + 60x^2 + 474x + C \).

Step 4: Evaluating the Definite Integral

Next, evaluate this from \( x=0 \) to \( x=10 \), i.e., \( \left[ \frac{14.2}{3}x^3 + 60x^2 + 474x \right]_{0}^{10} \). Calculate: \[ \frac{14.2}{3}(10)^3 + 60(10)^2 + 474(10) - \left(\frac{14.2}{3}(0)^3 + 60(0)^2 + 474(0)\right) \].

Step 5: Solving and Computing the Average

Calculate the values: \( \frac{14.2}{3}(1000) + 60(100) + 4740 = 4733.33 + 6000 + 4740 = 15473.33 \). So the definite integral result is \( 15473.33 \). Then, compute the average: \( \frac{1}{10} \times 15473.33 = 1547.33 \).

Step 6: Conclusion

The average annual revenue from 2013 to 2023 is approximately \( 1547.33 \) million dollars.

Key Concepts

Definite Integral

Definite integrals are a fundamental concept in calculus, helping us evaluate the total accumulation of quantities over a certain interval. They are represented by the integral symbol with limits of integration, typically \( \int_{a}^{b} f(x) \, dx \). This operation allows for the calculation of areas under a curve from points \(a\) to \(b\), where \(f(x)\) is a function of \(x\).

In our case, the definite integral is used to determine the total revenue predicted by the polynomial function \(14.2x^2 + 120x + 474\), over a 10-year period, from 2013 to 2023. Here, \(x\) ranges from 0 to 10, with each increment representing a year. By finding the value of this integral, we get the total predicted revenue over these years before averaging it to find the annual value.

Understanding how definite integrals work is crucial in contexts like business and physics, as they provide comprehensive summaries of continuous quantities across defined intervals.

Revenue Prediction

Revenue prediction involves forecasting future income, which is essential for businesses planning long-term strategies. Here, mathematical models enable companies to expect potential earnings, aiding in decision-making.

In the given exercise, the revenue prediction model is based on a polynomial function of \(14.2x^2 + 120x + 474\), which models the revenue in millions over time. By using this function, we can approximate how much the company will earn annually from 2013 to 2023 by averaging the calculated sums across these years.

• Revenue prediction is crucial for budgeting and investment.
• Makes strategic planning informed and less risky.
• Helps identify trends and address potential issues proactively.

Through this model, companies can navigate economic uncertainties while having clearer financial expectations extending through the next decade.

Mathematical Modeling

Mathematical modeling is a process of creating a mathematical representation of a real-world situation. It helps in understanding and solving complex problems by using equations to depict scenarios. In this case, we use a polynomial function to model the revenue.

Mathematical models like this one provide valuable insights into how different variables could affect outcomes. In our exercise, the polynomial function assigned includes terms for quadratic and linear effects relating to time. Such models give analysts and decision-makers tools for:

• Visualizing potential growth or decline in revenue.
• Forecasting future operations based on present data.
• Testing various hypothetical scenarios to gauge impacts.

With the aid of mathematical models, businesses can simulate and evaluate strategies over varying economic landscapes, leading to more robust planning and competitive advantage.

Polynomial Function

A polynomial function is a mathematical expression that consists of variables and coefficients, arranged in terms of powers and constants. Polynomials are versatile due to their ability to model various types of data across numerous contexts.

In the exercise, the polynomial \(14.2x^2 + 120x + 474\) helps predict revenue over time. It comprises three distinct components:

• The term \(14.2x^2\) represents changes that accelerate over time, possibly modeling compounding growth.
• The term \(120x\) provides a linear increase, suggesting consistent yearly revenue changes.
• The constant term \(474\) denotes an initial revenue base independent of time.

Polynomials in such applications act as predictive toolkits and offer a graspable structure for evaluating changes over a specified period. By understanding each element of the polynomial, students and analysts alike can infer how each factor contributes to the overall model.