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.