Problem 86 - 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, VALUE, or TABLE.] c. Predict the global wind power generating capacity in the year \(2020 .\)

Step-by-Step Solution

Answer

The capacity is 294.4 thousand MW for 2015 and 477.4 thousand MW for 2020.

Step 1: Understanding the Equation

The given equation for global wind power generating capacity is \( y = 0.9x^2 - 3.9x + 12.4 \), where \( y \) represents the capacity in thousand megawatts, and \( x \) is the number of years after 1995. We will use this equation to graph the curve and make future predictions.

Step 1: Graphing the Curve

To graph the curve, create a table of values for \( x \) from 0 to 20, as we want to interpret the curve on the window [0, 20] for \( x \) and [0, 300] for \( y \). Calculate \( y \) for each of these values, and plot these points on the graph.

Step 2: Determine the 2015 Value

To find the wind power capacity in 2015, use \( x = 20 \) since 2015 is 20 years after 1995. Substitute \( x = 20 \) into the equation:\[y = 0.9(20)^2 - 3.9(20) + 12.4\]Calculate \( y \) to find the predicted capacity.

Step 3: Calculate 2015 Capacity

Calculate the value:\[y = 0.9(400) - 3.9(20) + 12.4 = 360 - 78 + 12.4 = 294.4\]The predicted global wind power capacity for 2015 is 294.4 thousand MW.

Step 4: Determine the 2020 Value

To predict the capacity for 2020, use \( x = 25 \) since 2020 is 25 years after 1995. Substitute \( x = 25 \) into the equation:\[y = 0.9(25)^2 - 3.9(25) + 12.4\]Calculate \( y \) to predict the capacity.

Step 5: Calculate 2020 Capacity

Calculate the value:\[y = 0.9(625) - 3.9(25) + 12.4 = 562.5 - 97.5 + 12.4 = 477.4\]The predicted global wind power capacity for 2020 is 477.4 thousand MW.

Key Concepts

Wind Energy

Wind energy is a key player in the realm of renewable resources. It harnesses the kinetic energy produced by wind and transforms it into electrical power. This is achieved through wind turbines which are designed to capture this energy efficiently. The utilization of wind energy is picking up pace worldwide, reflecting its potential in reducing our reliance on fossil fuels. Unlike traditional energy sources, wind energy is clean, with no greenhouse gas emissions, making it environmentally friendly.

Wind power has also become increasingly economically viable thanks to advancements in technology.
Its scalability means it can be deployed in various sizes, from small residential turbines to large wind farms.
Wind energy projects contribute to energy security, reducing the dependence on imported fuels, and bring economic benefits like job creation and community investments.

Being a sustainable resource, wind energy has the capability to play a significant role in addressing climate change, offering a promising alternative to conventional power sources.
As technology continues to evolve, the efficiency and availability of wind energy are expected to rise, meeting the growing global energy demands.

Quadratic Equations

Quadratic equations are fundamental in algebra and applied mathematics. These are polynomial equations of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). In the context of our wind energy prediction, the given quadratic equation \( y = 0.9x^2 - 3.9x + 12.4 \) plays a crucial role.
The solutions to quadratic equations are typically found using various methods:

• Factoring: Expressing the quadratic in a product of linear factors, when possible.
• Quadratic Formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
• Completing the Square: Rewriting the equation as a perfect square.
• Graphing: Visualizing the equation to find the roots, vertex, and axis of symmetry.

In applications like predictions, quadratic equations help model real-world situations where relationships are not linear.
This makes them essential tools in areas ranging from physics to finance.

Graphing Functions

Graphing functions is a powerful way to visualize relationships between variables and understand patterns in data. For a quadratic function like \( y = 0.9x^2 - 3.9x + 12.4 \), graphing helps us see the parabolic shape typical of such equations.

When graphing quadratic equations, there are a few important features to identify:

• Vertex: The highest or lowest point on the graph. This can be found using \( x = -\frac{b}{2a} \).
• Axis of Symmetry: A vertical line that passes through the vertex, helping to divide the parabola into two mirror images.
• Intercepts: Points where the graph crosses the axes. The y-intercept is found by setting \( x = 0 \), and the x-intercepts (if any) are the solutions of the equation.

In practical applications, graphing enables us to predict future outcomes by extrapolating the curve beyond known data points.
In our example, tracing the quadratic graph allows us to predict the future capacity of wind energy, illustrating its growth over time.
This visual representation is integral for scientists and engineers who use these models to make informed decisions and strategic planning.