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The problem asks to calculate the extrapolated weight of the puppy after 3 years using the quadratic model \( w(t) = 0.002t^2 + 4.156t - 1.886 \), where \( t \) is the time in months and \( w(t) \) is the weight in pounds.

### Step 1: Convert 3 years to months.
Since 1 year has 12 months, 3 years will be:
\[
t = 3 \times 12 = 36 \text{ months}.
\]

### Step 2: Substitute \( t = 36 \) into the given quadratic equation.
\[
w(36) = 0.002(36)^2 + 4.156(36) - 1.886
\]

### Step 3: Perform the calculations.
First, calculate \( 36^2 \):
\[
36^2 = 1296
\]

Now substitute:
\[
w(36) = 0.002 \times 1296 + 4.156 \times 36 - 1.886
\]
\[
w(36) = 2.592 + 149.616 - 1.886
\]
\[
w(36) = 150.322 \text{ pounds}.
\]

### Step 4: Round the answer to the nearest pound.
\[
w(36) \approx 150 \text{ pounds}.
\]

Thus, the extrapolated weight of the puppy after 3 years is **150 pounds**.

Would you like further details or have any questions?

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Here are 5 related questions to deepen your understanding:

1. How does the quadratic model differ from a linear model for predicting growth?
2. Why is extrapolation potentially less accurate than interpolation?
3. What would happen to the weight prediction if you used a higher degree polynomial for modeling?
4. How would you check if this model is valid for more than 36 months?
5. Can you derive an equation that models weight based on the data table instead of using the given quadratic model?

**Tip**: When using extrapolation, be mindful that predictions far beyond the given data range may not be accurate, especially if the growth model isn't perfect for long-term trends.

James bought a puppy and has been tracking its weight at the end of each month. Using the quadratic function w(t) = 0.002t^2 + 4.156t - 1.886, what is the extrapolated weight of the dog after 3 years? Round your answer to the nearest pound.

Learn how to use a quadratic function to predict the weight of a puppy after 3 years. This problem covers core algebra concepts, including quadratic equations and extrapolation. Based on the function w(t) = 0.002t^2 + 4.156t - 1.886, the extrapolated weight of the puppy is calculated to be approximately 150 pounds.

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