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The image shows a table and some questions related to the capacity (in percentage) at which U.S. nuclear power plants were working at different years, and a quadratic function that models the data.

### Given Table of Capacity:
- **1975**: 56%
- **1980**: 59%
- **1985**: 58%
- **1990**: 70%
- **1995**: 76%
- **2000**: 88%
- **2004**: 89%

### Function provided for the model:
\[
f(t) = 0.027t^2 + 0.216t + 53.296
\]
Where:
- \( t \) is the number of years since 1970.
- \( f(t) \) is the capacity (in percent).

### Steps for solving:

1. **Graphing and Checking Fit:**
   - The model function provided is quadratic, and the scatter points come from the table. Based on the closeness of the data points to the model curve, you can decide if the function is a good fit for the data.
   - Looking at the equation, it captures the rising trend in the percentages (since it is a quadratic model with a positive leading coefficient).

2. **Estimate of Capacity in 2012:**
   - For 2012, we calculate \( t = 2012 - 1970 = 42 \) years since 1970.
   - Substitute \( t = 42 \) into the model function \( f(t) \).

3. **Predicting Full Capacity (100%):**
   - To predict when the capacity will reach 100%, set \( f(t) = 100 \) and solve for \( t \).

Let’s first calculate the required values using the model function.

### Estimate the capacity in 2012:

\[
f(42) = 0.027(42)^2 + 0.216(42) + 53.296
\]

### Predict when nuclear plants will reach 100% capacity:
Set \( f(t) = 100 \):
\[
0.027t^2 + 0.216t + 53.296 = 100
\]
Solve for \( t \).

I'll compute these values for you.### Results:

1. **Estimated capacity in 2012**: 
   The model predicts the capacity at which U.S. nuclear power plants were working in 2012 is approximately **110%**, which is unrealistic as capacities cannot exceed 100%. This suggests the model may not be perfectly accurate beyond certain years.

2. **Prediction for full (100%) capacity**: 
   The model predicts that U.S. nuclear power plants would have reached full capacity around **37.78 years** after 1970, which corresponds to the year **2008**.

Would you like further clarification or details on these calculations?

### Related Questions:
1. How does a quadratic model differ from a linear model in fitting this data?
2. Why might the capacity model predict values greater than 100%?
3. Could a logistic model be a better fit for capacity data that has an upper limit like 100%?
4. How sensitive is the model's prediction of full capacity to small changes in the coefficients?
5. How would you adjust the model to better fit future predictions?

### Tip:
When predicting future trends with a quadratic model, be cautious about applying it far beyond the data range—especially if there's a natural limit, like 100%.

The capacities at which U.S. nuclear power plants are working are shown in the table for various years. A model of the situation is given by the quadratic function f(t) = 0.027t^2 + 0.216t + 53.296, where t is the number of years since 1970. Estimate capacity in 2012 and predict when nuclear power plants will be working at full (100%) capacity.

Learn how to estimate the capacity of U.S. nuclear power plants using a quadratic function model f(t) = 0.027t^2 + 0.216t + 53.296. Predict when full capacity (100%) will be reached and analyze the function fit with historical data. Suitable for high school students studying quadratic functions.

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