Problem 111 - Calculus Volume 1

Question

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 percentage output in 2015 is about 95.916%.

Step 1: Identify the value of t for 2015

Since the function uses \( t = 0 \) to represent the beginning of the year 2000, calculate \( t \) for the year 2015 by subtracting 2000 from 2015. Thus, \( t = 2015 - 2000 = 15 \).

Step 2: Substitute t into the function

Replace \( t \) in the function \( P(t) = 1.8576t + 68.052 \) with the value from Step 1. Therefore, evaluate \( P(15) = 1.8576 \times 15 + 68.052 \).

Step 3: Perform the calculation

Evaluate the expression from Step 2. First, multiply: \( 1.8576 \times 15 = 27.864 \). Then add: \( 27.864 + 68.052 = 95.916 \).

Step 4: Interpret the result

The value \( P(15) = 95.916 \) represents the nuclear power output as a percentage of total capacity for the year 2015. Therefore, the model predicts that the output will be approximately 95.916%.

Key Concepts

Modeling Real-World Problems

Modeling real-world problems involves translating a situation from everyday life into a mathematical equation. In this exercise, we used a linear function to model the output of nuclear power plants over time. This means representing how the power output percentage changes from year to year.

• The model is given as a linear function: \( P(t) = 1.8576t + 68.052 \), where \( t \) is time in years since 2000.
• The coefficients and constants in the function have specific meanings. The 1.8576 represents the yearly increase in output percent, while 68.052 signifies the percentage output at the start of 2000.

Using such a mathematical model helps project future trends based on past data, providing valuable insights into power plant operations.

Function Evaluation

Function evaluation is the process of substituting a specific value for the variable in a given function. For this exercise, we need to evaluate the function to predict nuclear power plant output for the year 2015.
The key steps include:

• First, determine the value of \( t \) by counting the number of years from the starting point (the year 2000) to your year of interest (2015). Here, \( t = 15 \).
• Next, substitute \( t = 15 \) into the function \( P(t) = 1.8576t + 68.052 \).
• Finally, perform the arithmetic operations to find the percentage output.

Following these steps enables a clear understanding of how changes in time affect outcomes, a critical skill for solving real-world problems with functions.

Predictive Modeling

Predictive modeling uses data to forecast outcomes. In this context, we forecast the percentage of nuclear power output for future years. The linear function used here is a simple example.
Our prediction involved these steps:

• Identifying the function components: a slope of 1.8576 and an intercept of 68.052.
• Using these components to generate forecasts: by plugging different values of \( t \), we predict output data for future years.

Such models are powerful as they can help industry specialists plan resource allocation, anticipate power production needs, and implement policy changes. Be aware that while useful, models are simplifications and actual results may vary due to unforeseen factors.

Step-by-Step Solution

Breaking down a problem into smaller, manageable steps is critical for understanding. Here’s how the calculation process works for the given model:

• Step 1: Calculate \( t \). For the year 2015, \( t = 15 \).
• Step 2: Substitute into the function. Replace \( t \) with 15, setting up the expression \( P(15) = 1.8576 \times 15 + 68.052 \).
• Step 3: Simplify the expression by performing multiplication first: \( 1.8576 \times 15 = 27.864 \).
• Add the product to the constant: \( 27.864 + 68.052 = 95.916 \).
• Step 4: Interpret the result. The power output in 2015 is predicted to be 95.916% of capacity.

Step-by-step solutions make it easier to track your process and identify where errors might occur. They build a strong foundation for tackling more complex problems in the future.