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Let’s solve the given problems step by step:

### (a) Setting up the Quadratic Model
We need to find a quadratic polynomial \( p(z) = a_0 + a_1 z + a_2 z^2 \) that fits the data for the years 2000, 2002, and 2004. To simplify calculations, we use the variable transformation \( z = x - 2000 \), where \( x \) is the year.

The data points are:
- \( z = 0 \) (year 2000), \( p(0) = 62 \)
- \( z = 2 \) (year 2002), \( p(2) = 70 \)
- \( z = 4 \) (year 2004), \( p(4) = 94 \)

We need to find \( a_0 \), \( a_1 \), and \( a_2 \) that satisfy the system of linear equations based on these conditions:
1. \( p(0) = a_0 = 62 \)
2. \( p(2) = a_0 + 2a_1 + 4a_2 = 70 \)
3. \( p(4) = a_0 + 4a_1 + 16a_2 = 94 \)

### (b) Solving the System using Gaussian Elimination
Let's substitute \( a_0 = 62 \) into the second and third equations:
1. \( 62 + 2a_1 + 4a_2 = 70 \) ⟹ \( 2a_1 + 4a_2 = 8 \)
2. \( 62 + 4a_1 + 16a_2 = 94 \) ⟹ \( 4a_1 + 16a_2 = 32 \)

Dividing the first equation by 2 gives:
- \( a_1 + 2a_2 = 4 \)

Next, we subtract this equation from the second equation:
- \( (4a_1 + 16a_2) - (2a_1 + 8a_2) = 32 - 8 \)
- \( 2a_1 + 8a_2 = 24 \)

Dividing by 2 gives:
- \( a_1 + 4a_2 = 12 \)

Now, subtract \( a_1 + 2a_2 = 4 \) from \( a_1 + 4a_2 = 12 \):
- \( (a_1 + 4a_2) - (a_1 + 2a_2) = 12 - 4 \)
- \( 2a_2 = 8 \), thus \( a_2 = 4 \)

Substitute \( a_2 = 4 \) into \( a_1 + 2a_2 = 4 \):
- \( a_1 + 8 = 4 \), hence \( a_1 = -4 \)

The interpolating polynomial is:
\[ p(z) = 62 - 4z + 4z^2 \]

### (c) Evaluating the Cubic Model for the Year 2003
The given cubic model is:
\[ p(z) = 62 - 32z + 26z^2 - 4z^3 \]
For 2003, \( z = 3 \):
\[ p(3) = 62 - 32(3) + 26(3^2) - 4(3^3) \]
\[ = 62 - 96 + 234 - 108 \]
\[ = 62 + 234 - 204 \]
\[ = 92 \]

The model predicts sales of 92 thousand euros for 2003, which aligns reasonably well with the observed trend. 

Would you like more details, or have any questions?

### Related Questions:
1. How would the solution change if we used different data points (e.g., 2001, 2003, 2005)?
2. Can we use a different polynomial degree for a better fit, such as linear or quartic?
3. How does the choice of \( z = x - 2000 \) affect the calculations?
4. What are the limitations of using polynomial interpolation for forecasting?
5. How would we solve this problem using numerical methods or computational tools?

### Tip:
When interpolating data, ensure that the chosen polynomial degree matches the data's behavior to avoid overfitting or underfitting.

As vendas (em milhares de euros) da loja 'Doces da Lu' de 2000 a 2005 são mostradas na tabela. Encontrar um modelo quadrático e cúbico que se ajustem aos dados e verificar a previsão para 2003.

Learn how to model sales data for 'Doces da Lu' from 2000 to 2005 using polynomial interpolation. This problem covers quadratic and cubic polynomial models, Gaussian elimination, and prediction analysis for the year 2003.

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