Problem 111 - Calculus Volume 1

Question

ITI The output (as a percent of total capacity) of nuclear power plants in the United States can be modeled by the function \(P(t)=1.8576 t+68.052,\) where \(t\) is time in years and \(t=0\) corresponds to the beginning of 2000 . Use the model to predict the percentage output in 2015 .

Step-by-Step Solution

Answer

The predicted output in 2015 is 95.916% of total capacity.

Step 1: Identify the Variable 't'

To predict the output for 2015, we must first calculate the value of \( t \). Since \( t = 0 \) corresponds to the year 2000, the year 2015 corresponds to \( t = 2015 - 2000 = 15 \).

Step 2: Substitute 't' into the Function

Now that we know \( t = 15 \), we substitute it into the function \( P(t) = 1.8576t + 68.052 \) to find the output percentage for that year.

Step 3: Calculate the Output Percentage

Replace \( t \) in the equation with 15: \( P(15) = 1.8576 \times 15 + 68.052 \).Calculate: \( P(15) = 27.864 + 68.052 = 95.916 \).

Step 4: Conclusion

The model predicts that in 2015, the output of U.S. nuclear power plants was 95.916% of total capacity.

Key Concepts

Mathematical Modeling

Mathematical modeling is a powerful tool that uses mathematical expressions to represent real-world scenarios. In this exercise, we use the linear function \( P(t) = 1.8576t + 68.052 \) to describe the output of nuclear power plants in the U.S. over time. The function is based on the assumption that the output increases linearly with time. Here, \( P(t) \) represents the percentage output of capacity, and \( t \) denotes the number of years since 2000. Such models help in understanding trends and making predictions about future outcomes.

• Linear functions consist of a constant rate of change.
• They follow the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
• In our model, \( 1.8576 \) is the slope, indicating the increase in output percentage per year.

This mathematical model simplifies the complex operation of power plants into manageable predictions, allowing for planning and analysis.

Calculating Percentages

When it comes to percentages, they are a metric to express a number as a fraction of 100. In our exercise, we want to find the nuclear power output as a percentage of its total capacity for the year 2015. To achieve this, we substitute \( t = 15 \) into our function since 2015 is 15 years past 2000.

First, multiply the coefficient \( 1.8576 \) by 15, which gives \( 27.864 \). Then, add this product to the constant term \( 68.052 \) in the linear function.

• The addition yields a total of \( 95.916 \).
• This result indicates that the nuclear power plants would operate at 95.916% of their total capacity in 2015.

Calculating percentages in this way allows us to assess performance as it relates to the total possible output, offering a clear metric for efficiency.

Predictive Analysis

Predictive analysis involves using historical data to forecast future events or trends. By applying the linear model \( P(t) = 1.8576t + 68.052 \), we leverage mathematical calculations to predict the future performance of nuclear power plants. This method contains several valuable steps:

• Identify the year you desire to predict, deduct from the reference year (2000) to determine the value of \( t \).
• Substitute \( t \) into the established function.
• Compute to find the outcome, which offers a peek into possible future outputs.

The process of predictive analysis helps businesses, governments, and other entities to prepare and make informed decisions based on projected data trends. By predicting that nuclear power plants would operate at 95.916% capacity in 2015, we understand the trajectory of increased efficiency and can plan accordingly for energy needs and policies in the future.