**U.S. Nuclear Power Plant Capacity Over Time**The table below represents the capacities, in percentage, at which U.S. nuclear power plants have been operating over various years:| Year | Percent ||------|---------|| 1975 | 56 || 1980 | 59 || 1985 | 58 || 1990 | 70 || 1995 | 76 || 2000 | 88 || 2004 | 89 |**Mathematical Model**Let $ f(t) $ be the capacity (in percent) at which U.S. nuclear power plants are working at $ t $ years since 1970. A model of the situation is given by:\[ f(t) = 0.027t^2 + 0.216t + 53.296 \]**Exercise: Data Visualization and Model Evaluation**Using a graphing calculator:1. Plot the graph of the function $ f(t) $.2. Plot the scattergram using the data points from the table.3. Evaluate: Does the model fit the data well?- $ \circ $ The function is not a good model for the data.- $ \circ $ The function is a good model for the data.**Prediction Task**Estimate at what capacity U.S. nuclear power plants were operating in 2010. ### Capacity of U.S. Nuclear Power Plants Over TimeThe table below shows the percentage capacity at which U.S. nuclear power plants have been operating at various years since 1970:| Year | Capacity (%) ||------|--------------|| 1990 | 70 || 1995 | 76 || 2000 | 88 || 2004 | 89 |#### Mathematical ModelLet $ f(t) $ represent the capacity (in percent) of U.S. nuclear power plants operating at $ t $ years since 1970. The model representing this situation is given by:\[ f(t) = 0.027t^2 + 0.216t + 53.296 \]#### Graphing and Analysis- Use a graphing calculator to plot the function $ f(t) $ and compare it with the scatter plot of the data provided in the table.- Evaluate the fit of the model to the data.**Options:**- The function is not a good model for the data.- The function is a good model for the data.#### Estimations and Predictions1. **Estimate the Capacity for 2010:** - Calculate the capacity percentage for the year 2010 using the model. Round your answer to the nearest whole percent.2. **Predict Full Capacity:** - Determine the year when U.S. nuclear power plants will operate at full (100%) capacity using the model.This exercise aids in understanding how mathematical modeling and calculations are used to predict trends and capacities in real-world scenarios like nuclear power plant operations.

Consider the given table. Year Percent 1975 56 1980 59 1985 58 1990 70 1995 76 2000 88 2004 89Now, use the graphing calculator to find the model. By using excel, the equation for the given table is y=0.0271x2-0.2165x+53.296.Replace x with t and rewrite the equation. y=0.0271t2-0.2165t+53.296 As, the calculated equation is same as the given equation. Thus, the function is good model for the data. (b) Now, calculate foe the year 2010. t=2010-1970=40Now, put 40 for t in the model and calculate the value. y40=0.0271402+0.216540+53.296=43.36+8.66+53.296=105.316 Thus, the capacity of nuclear plant in 2010 is 105.316%. (c) Now, calculate the time when the capacity of planet is 100%. 0.0271t2+0.2165t+53.296=1000.0271t2+0.2165t-46.704=0Now, use the graphing quadratic formula to find the root of the equation. t=-0.2165±0.21652-40.0271-46.70420.0271=-0.2165±5.1095858520.0271=-0.2165±2.260430.0542=37.71105,- 45.69998≈38,-46 Take the positive value of the equation. Thus, the capacity of planet will be 100% in 38 years (2008).

Math

Advanced Math

Let f(t) be the capacity (in percent) at which U.S. nuclear power plants are working at t years since 1970. A model of the situation is f(t) 0.027t2 + 0.216t + 53.296. %3D Use a graphing calculator to draw the graph of the model and, in the same viewing window, the scattergram of the data. Does the model fit the data well? O The function is not a good model for the data O The function is a good model for the data. Estimate at what capacity U. S. nuclear power plants were working in 2010. % Round to the nearest whole percent. Predict when U. S. nuclear power plants will be working at full (100%) capacity. Enter the year this occurs.

Let f(t) be the capacity (in percent) at which U.S. nuclear power plants are working at t years since 1970. A model of the situation is f(t) 0.027t2 + 0.216t + 53.296. %3D Use a graphing calculator to draw the graph of the model and, in the same viewing window, the scattergram of the data. Does the model fit the data well? O The function is not a good model for the data O The function is a good model for the data. Estimate at what capacity U. S. nuclear power plants were working in 2010. % Round to the nearest whole percent. Predict when U. S. nuclear power plants will be working at full (100%) capacity. Enter the year this occurs.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

**U.S. Nuclear Power Plant Capacity Over Time**

The table below represents the capacities, in percentage, at which U.S. nuclear power plants have been operating over various years:

| Year | Percent |
|------|---------|
| 1975 | 56 |
| 1980 | 59 |
| 1985 | 58 |
| 1990 | 70 |
| 1995 | 76 |
| 2000 | 88 |
| 2004 | 89 |

**Mathematical Model**

Let $ f(t) $ be the capacity (in percent) at which U.S. nuclear power plants are working at $ t $ years since 1970. A model of the situation is given by:

\[ f(t) = 0.027t^2 + 0.216t + 53.296 \]

**Exercise: Data Visualization and Model Evaluation**

Using a graphing calculator:

1. Plot the graph of the function $ f(t) $.
2. Plot the scattergram using the data points from the table.
3. Evaluate: Does the model fit the data well?

- $ \circ $ The function is not a good model for the data.
- $ \circ $ The function is a good model for the data.

**Prediction Task**

Estimate at what capacity U.S. nuclear power plants were operating in 2010.

### Capacity of U.S. Nuclear Power Plants Over Time

The table below shows the percentage capacity at which U.S. nuclear power plants have been operating at various years since 1970:

| Year | Capacity (%) |
|------|--------------|
| 1990 | 70 |
| 1995 | 76 |
| 2000 | 88 |
| 2004 | 89 |

#### Mathematical Model

Let $ f(t) $ represent the capacity (in percent) of U.S. nuclear power plants operating at $ t $ years since 1970. The model representing this situation is given by:

\[ f(t) = 0.027t^2 + 0.216t + 53.296 \]

#### Graphing and Analysis

- Use a graphing calculator to plot the function $ f(t) $ and compare it with the scatter plot of the data provided in the table.
- Evaluate the fit of the model to the data.

**Options:**
- The function is not a good model for the data.
- The function is a good model for the data.

#### Estimations and Predictions

1. **Estimate the Capacity for 2010:**
- Calculate the capacity percentage for the year 2010 using the model. Round your answer to the nearest whole percent.

2. **Predict Full Capacity:**
- Determine the year when U.S. nuclear power plants will operate at full (100%) capacity using the model.

This exercise aids in understanding how mathematical modeling and calculations are used to predict trends and capacities in real-world scenarios like nuclear power plant operations.

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