The per capita consumption of commercially produced fresh vegetables in a certain country from 1980 through 2000 was as shown in the accompanying table.
Per capita consumption of fresh vegetables
in a certain country
Year	Vegetable consumption, V
(pounds per person)
1980	147.1
1985	153
1990	164.1
1995	178.1
2000	201.7

(a) Find the function of the quadratic model that gives the per capita consumption of fresh vegetables in pounds per person, where t is the number of years since 1980, with data from
0 ≤ t ≤ 20.
 Examine the equation graphed on a scatter plot of the data. (Round all numerical values to three decimal places.)
V(t) =   

(b) Do you believe that the equation in part (a) is a good fit?
The model does not appear to be a good fit.
The model appears to be a good fit.    
This cannot be determined.

(c) The per capita consumption in 2001 had not yet been tabulated when the data in the table were published. What does the quadratic model give as the per capita consumption in 2001? (Round your answer to one decimal place.)
  pounds per person

(d) According to your model, in what year will consumption exceed 225 pounds per person.
 

Let's solve each part step by step, starting with part (a).

### Part (a) - Quadratic Model

The data gives the per capita consumption \( V(t) \) of fresh vegetables (in pounds per person) for different years, where \( t \) represents the number of years since 1980.

The given data points are:
- \( t = 0 \) (1980), \( V(0) = 147.1 \)
- \( t = 5 \) (1985), \( V(5) = 153 \)
- \( t = 10 \) (1990), \( V(10) = 164.1 \)
- \( t = 15 \) (1995), \( V(15) = 178.1 \)
- \( t = 20 \) (2000), \( V(20) = 201.7 \)

We are asked to find a quadratic model, meaning we want a function of the form:
\[
V(t) = at^2 + bt + c
\]
where \( a \), \( b \), and \( c \) are constants to be determined by fitting the data.

We can solve this using a least-squares regression method or by solving a system of equations using the given data points. Let me calculate the values of \( a \), \( b \), and \( c \) first.

### Step 1: Fit the quadratic model.

I'll compute the coefficients for the quadratic function \( V(t) = at^2 + bt + c \) using the data points provided.The quadratic model for the per capita consumption of fresh vegetables is:

\[
V(t) = 0.109t^2 + 0.497t + 147.411
\]

where \( t \) is the number of years since 1980.

### Part (b) - Is the model a good fit?

We can evaluate the goodness of fit by checking how well the quadratic model matches the data points. Since we used least-squares regression, the model minimizes the error between the predicted values and the actual data points. Based on this, the model **appears to be a good fit**.

### Part (c) - Prediction for 2001

To find the per capita consumption in 2001, we need to evaluate \( V(t) \) for \( t = 21 \), since 2001 is 21 years after 1980.

\[
V(21) = 0.109(21)^2 + 0.497(21) + 147.411
\]

Let me calculate that value.The quadratic model predicts that the per capita consumption of fresh vegetables in 2001 would be approximately **206.1 pounds per person**.

### Part (d) - Year when consumption exceeds 225 pounds

To determine when the per capita consumption exceeds 225 pounds, we solve the inequality:

\[
0.109t^2 + 0.497t + 147.411 > 225
\]

Let's solve for \( t \).The quadratic equation predicts that per capita vegetable consumption will exceed 225 pounds in the year corresponding to \( t \approx 24.45 \). Since \( t \) represents the number of years since 1980, this implies the consumption will exceed 225 pounds in **the year 2004**.

### Summary:
- **(a)** The quadratic model is \( V(t) = 0.109t^2 + 0.497t + 147.411 \).
- **(b)** The model appears to be a good fit.
- **(c)** Predicted per capita consumption in 2001 is approximately **206.1 pounds per person**.
- **(d)** Per capita consumption will exceed 225 pounds in **2004**.

Would you like further details or clarification on any part of this solution?

Here are 5 related questions to explore:
1. How would a linear model compare in accuracy to the quadratic model?
2. How is least-squares regression used to find the best-fit curve?
3. Can a cubic model provide an even better fit than the quadratic model?
4. What are the limitations of using a quadratic model for long-term predictions?
5. How do outliers in data affect the accuracy of regression models?

**Tip:** When fitting models to data, it is crucial to evaluate the model's fit and not just rely on equations—this helps ensure accurate predictions.

The per capita consumption of commercially produced fresh vegetables in a certain country from 1980 through 2000 was as shown in the accompanying table. (a) Find the function of the quadratic model that gives the per capita consumption of fresh vegetables in pounds per person, where t is the number of years since 1980, with data from 0 ≤ t ≤ 20. (b) Do you believe that the equation in part (a) is a good fit? (c) The per capita consumption in 2001 had not yet been tabulated when the data in the table were published. What does the quadratic model give as the per capita consumption in 2001? (d) According to your model, in what year will consumption exceed 225 pounds per person.

Explore how to derive a quadratic model for the per capita consumption of fresh vegetables in a country between 1980 and 2000. Using a least-squares regression, we find the function V(t) = 0.109t^2 + 0.497t + 147.411 to model the data and predict future consumption trends, including the year consumption will exceed 225 pounds per person.

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