Problem 86 - Brief Applied Calculus

Question

ENVIRONMENTAL SCIENCE: Wind Energy The use of wind power is growing rapidly after a slow start, especially in Europe, where it is seen as an efficient and renewable source of energy. Global wind power generating capacity for the years 1996 to 2008 is given approximately by \(y=0.9 x^{2}-3.9 x+12.4\) thousand megawatts (MW), where \(x\) is the number of years after 1995 . (One megawatt would supply the electrical needs of approximately 100 homes). a. Graph this curve on the window \([0,20]\) by \([0,300]\). b. Use this curve to predict the global wind power generating capacity in the year \(2015 .\) [Hint: Which \(x\) -value corresponds to \(2015 ?\) Then use TRACE, EVALUATE, or TABLE.] c. Predict the global wind power generating capacity in the year \(2020 .\)

Step-by-Step Solution

Answer

In 2015, it is 294.4 thousand MW; in 2020, it is 477.4 thousand MW.

Step 1: Understanding the Equation

The equation given is \( y = 0.9x^2 - 3.9x + 12.4 \), where \( y \) represents the global wind power generating capacity in thousand megawatts, and \( x \) is the number of years after 1995. Our task is to predict values for specific years, so first, we need to convert years to their corresponding \( x \) values.

Step 2: Convert Year to x-value

For any given year, the corresponding \( x \) value is found by subtracting 1995 from the year. Therefore, for 2015, \( x = 2015 - 1995 = 20 \). For 2020, \( x = 2020 - 1995 = 25 \).

Step 3: Graph the Curve

Graph the curve \( y = 0.9x^2 - 3.9x + 12.4 \) in the window \([0,20]\) for the x-axis and \([0,300]\) for the y-axis. The graph will show you how the global wind power generating capacity changes over the years from 1996 to 2015.

Step 4: Predict for 2015 (x = 20)

Substitute \( x = 20 \) into the equation to find \( y \). Calculate: \[ y = 0.9(20)^2 - 3.9(20) + 12.4 = 0.9 imes 400 - 78 + 12.4 = 360 - 78 + 12.4 = 294.4 \]. The prediction for 2015 is 294.4 thousand MW.

Step 5: Predict for 2020 (x = 25)

Substitute \( x = 25 \) into the equation to find \( y \). Calculate: \[ y = 0.9(25)^2 - 3.9(25) + 12.4 = 0.9 imes 625 - 97.5 + 12.4 = 562.5 - 97.5 + 12.4 = 477.4 \]. The prediction for 2020 is 477.4 thousand MW.

Key Concepts

Understanding Quadratic Equations

Quadratic equations are an essential part of mathematics, often appearing in diverse fields such as physics, engineering, and economics. These equations are generally in the form of \[ ax^2 + bx + c = 0 \]where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). In this format, the term \( ax^2 \) is the quadratic term, \( bx \) is the linear term, and \( c \) is the constant term.

• The graph of a quadratic equation is a parabola.
• Parabolas can open upwards or downwards depending on the sign of \( a \).
• If \( a > 0 \), the parabola opens upwards. If \( a < 0 \), it opens downwards.

In the case of wind energy, the equation \[ y = 0.9x^2 - 3.9x + 12.4 \]models the global wind power generating capacity. The positive \( a \) value causes the parabola to open upwards, indicating that wind power capacity is expected to increase over the years, given the assumptions don't change drastically.

Graphing Functions

Graphing functions allows us to visualize relationships between variables that mathematics describes. For quadratic functions, the graph is a parabola. Here is how you can effectively graph one:

• Identify the vertex, the highest or lowest point of the parabola. The vertex of a standard quadratic \( y = ax^2 + bx + c \) is located at \( x = \frac{-b}{2a} \).
• Calculate the y-intercept, which is the point \((0, c)\).
• Determine the x-intercepts by solving the equation \( ax^2 + bx + c = 0 \).

These intercepts will be pivotal as they tell you where the graph crosses the x and y axes. In our specific context, graphing the wind capacity over the specific window \([0, 20]\) for x and \([0, 300]\) for y displayed the trend that the power capacity is on the rise, affirming that wind is becoming a more significant energy source. Always use graphing tools or software to gain precise figures when tackling practical problems.

Predictive Modeling with Mathematics

Predictive modeling is where mathematics meets real-world applications. It involves using mathematical equations to make predictions about future events based on past data.This technique is widely used in multiple fields such as finance for stock predictions, epidemiology for disease spread, and renewable energy for predicting capacities.

• With quadratic equations like \[ y = 0.9x^2 - 3.9x + 12.4 \],we can calculate the expected global wind power capacity for given years by substituting \( x \) values.
• For example, by setting \( x = 20 \)for the year 2015, our calculated result is 294.4 thousand MW. Similarly, substituting \( x = 25 \)for 2020, we predict 477.4 thousand MW.
• This shows the escalating trend of wind energy generation capacity over time. Predictive modeling enables planning and decision-making aligned with future energy needs and environmental goals.

Harness this tool for strategic forecasting and resource management.