Problem 1 - Applied Calculus

Question

World wind energy generating \(^{59}\) capacity, \(W\), was 18,000 megawatts in 2000 and has been increasing at a continuous rate of approximately \(27 \%\) per year. Assume this rate continues. (a) Give a formula for \(W\), in megawatts, as a function of time, \(t\), in years since 2000 . (b) When is wind capacity predicted to pass 250,000 megawatts?

Step-by-Step Solution

Answer

W(t) = 18,000e^{0.27t}; capacity exceeds 250,000 MW around 2010.

Step 1: Understand the Problem

We have an initial wind energy generation capacity of 18,000 megawatts in the year 2000. The capacity increases at a continuous rate of 27% per year, and we need to find a formula for this growth and determine when it will exceed 250,000 megawatts.

Step 2: Derive the Exponential Growth Formula

We use the formula for continuous growth: \( W(t) = W_0 e^{rt} \), where \( W_0 = 18,000 \) MW is the initial capacity, \( r = 0.27 \) is the continuous growth rate, and \( t \) is the time in years since 2000. Thus, the formula becomes: \[ W(t) = 18,000 e^{0.27t} \]

Step 3: Set Up the Equation for Capacity Exceeding 250,000 MW

To find when the capacity exceeds 250,000 MW, we set up the equation: \[ 18,000 e^{0.27t} = 250,000 \] to solve for \( t \).

Step 4: Solve for Time, \( t \)

First, divide both sides by 18,000: \[ e^{0.27t} = \frac{250,000}{18,000} \] Calculate the right side: \[ e^{0.27t} = 13.8889 \] Take the natural logarithm of both sides: \[ 0.27t = \ln(13.8889) \] Finally, divide by 0.27 to solve for \( t \): \[ t = \frac{\ln(13.8889)}{0.27} \]

Step 5: Calculate the Solution for \( t \)

Calculate \( \ln(13.8889) \) which is approximately 2.6311. Then compute \( t \): \[ t = \frac{2.6311}{0.27} \approx 9.74 \] This means the capacity will exceed 250,000 MW approximately 9.74 years after 2000, which corresponds to the year 2009 or early 2010.

Key Concepts

Continuous Growth Rate

Continuous growth rate refers to a situation where a quantity increases continuously over time at a steady percentage rate. In our example, wind energy generation grows by 27% per year. This rate is not just in whole yearly jumps; it is viewed as happening smoothly and continuously. Such growth can be modeled mathematically using an exponential function.

This type of growth is commonly seen in populations, investments, and various natural processes. It emphasizes compounding, where each year's growth builds upon the last, leading to rapid increases over time. To find the amount after certain years, we use the exponential growth formula:

• \( W(t) = W_0 e^{rt} \)

Here, \( W_0 \) is the initial amount, \( r \) is the growth rate, and \( t \) is time.

Understanding continuous growth helps in planning resources and understanding future projections, essential for sectors like renewable energy.

Wind Energy Generation

Wind energy generation, a cleaner and renewable energy source, depends on the efficient conversion of wind into electricity. It has been gaining momentum due to its low environmental impact and sustainability. The initial capacity in our exercise is based on empirical data from the year 2000, where the capacity was 18,000 megawatts.

The growth in wind energy generation is catalyzed by technological advances and increased investment. This growth rate, modeled continuously, illustrates the ever-increasing role wind energy plays in the global energy landscape. Harnessing wind effectively requires installations like wind farms, equipped with turbines converting kinetic wind energy into electricity.

As the world seeks renewable energy to reduce carbon footprints, understanding the growth potential of wind energy is crucial for future infrastructure planning and environmental policies.

Exponential Functions

Exponential functions are powerful mathematical tools used to describe growth and decay processes in various fields. They are particularly important in modeling situations with continuous, constant percentage changes over time, such as with interest rates or population growth.

In the context of wind energy, the exponential function \( W(t) = 18,000 e^{0.27t} \) represents how capacity increases over time. Here, \( e \) is the base of the natural logarithm, approximately equal to 2.71828.

Exponential functions have distinctive features:

• Non-linear growth: The rate increases over time, leading to faster rises.
• Continuous compounding: Allows us to calculate growth not just yearly, but at any continuous point in time.
• Predictive power: Helps determine future values, such as when the capacity will surpass 250,000 megawatts.

Understanding exponential functions enables accurate modeling and forecasting, making them indispensable in scientific and financial contexts.