Problem 10 - Applied Calculus

Question

World wind energy generating is 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 World wind energy generating is 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

Step-by-Step Solution

Answer

The formula is \(W(t) = 18000 \cdot e^{0.27t}\).

Step 1: Understand the Problem

We need to express the world wind energy generating capacity, \(W\), as a function of time \(t\), where \(t\) is the number of years since the year 2000. This requires using the given initial capacity of 18,000 megawatts and the continuous growth rate of 27% per year.

Step 2: Use the Formula for Exponential Growth

The formula for continuous exponential growth is given by \(W(t) = W_0 \cdot e^{rt}\) where \(W_0\) is the initial amount (18,000 megawatts), \(r\) is the growth rate as a decimal (0.27), and \(t\) is the time in years since the start year (2000).

Step 3: Substitute the Known Values into the Formula

Substitute \(W_0 = 18000\) and \(r = 0.27\) into the exponential growth formula:\[W(t) = 18000 \cdot e^{0.27t}\]

Step 4: Verify the Formula

Check the formula by ensuring that with \(t=0\), \(W(t)\) returns the initial capacity of 18,000 megawatts. For \(t=0\), \(W(0) = 18000 \cdot e^{0} = 18000\). This confirms the formula is accurate.

Key Concepts

Continuous Growth Rate

The concept of continuous growth rate is pivotal when analyzing situations where something expands consistently over time. In the context of wind energy capacity, a continuous growth rate indicates a consistent percentage increase per annum. This means that every year, the wind energy capacity grows by a certain percentage of the current capacity, not the original or base amount.

For wind energy, a growth rate of 27% was given. This translates to an increase of 27% of the current year's capacity added onto itself each year. This is a higher rate than you might see in more stable investments or industries, illustrating how rapidly wind energy generation is expanding. Continuous growth ensures that the gains are compounded, meaning each new year's growth builds on the gains of previous years.

• Continuous growth is typically expressed as an annual percentage rate.
• It's considered an ideal assumption for modeling to understand potential future gains.
• In practice, it shows how a given base amount grows exponentially over time.

Wind Energy Capacity

Wind energy capacity refers to the maximum amount of electrical power that wind energy systems, like wind turbines, are designed to produce. In the year 2000, the world wind energy generating capacity was 18,000 megawatts. This figure serves as the baseline from which further exponential growth is considered.

Understanding wind energy capacity is crucial for predicting future energy resources and sustainability. The initial figure provided acts as a critical starting point for predicting future capabilities using exponential growth models.

• The baseline figure of 18,000 megawatts provides a foundation for future forecasting.
• Capacity growth directly impacts energy policy and investment decisions.

Exponential Growth Formula

The exponential growth formula is a mathematical model used to demonstrate how quantities grow at a rate proportional to their current value. In this exercise, the formula is used to describe how the wind energy capacity grows over time. The general form of the formula for continuous exponential growth is:\[W(t) = W_0 \cdot e^{rt}\]Where:

• \(W(t)\) is the capacity at time \(t\).
• \(W_0\) is the initial capacity (18,000 megawatts in this case).
• \(r\) is the continuous growth rate as a decimal (0.27).
• \(t\) is time in years since the baseline year (2000).

This formula demonstrates how to project future capacities based on the given growth rate. For example, it can forecast how much capacity will grow after a certain number of years from the initial year.

The use of the exponential growth formula demonstrates the power of compounding, as each year, the growth adds to the existing total, making the potential increase much more significant over long periods. This is particularly useful for predicting the future scale and impact of emerging energies like wind power.