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The problem involves a quadratic model representing the weight \( w(m) \) of a baby as a function of their age in months \( m \):

\[
w(m) = -0.042m^2 + 1.75m + 8
\]

Let's break down each part:

### Part (a): Practical Interpretation of the 8
The constant term \( 8 \) in the equation represents the weight of the baby at \( m = 0 \) months, or at birth. This means the baby’s approximate weight at birth is 8 pounds.

### Part (b): Approximate Weight of a Baby at 3 Months Old
To find the weight of a 3-month-old baby, substitute \( m = 3 \) into the equation:

\[
w(3) = -0.042(3)^2 + 1.75(3) + 8
\]

First, calculate each term:
\[
-0.042(9) = -0.378
\]
\[
1.75(3) = 5.25
\]

Now, sum them up:
\[
w(3) = -0.378 + 5.25 + 8 = 12.872
\]

So, the approximate weight of the baby at 3 months old is **12.87 pounds**.

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Would you like further details or have any other questions?

Here are some related questions to extend this:
1. How does the coefficient of \( m^2 \) affect the weight over time?
2. What would be the weight at 6 months?
3. How can we determine the month when the baby's weight is at a maximum or minimum?
4. How does the model behave for values of \( m \) beyond 12 months?
5. Can this quadratic model be used for other types of growth prediction?

**Tip**: When working with quadratic models, the coefficient of \( m^2 \) determines the curvature of the graph, impacting whether the function increases or decreases after a certain point.

The weight of a baby during the first year of life is roughly approximated by a quadratic function of time. At month m, the weight in pounds is approximated by w(m) = -0.042m^2 + 1.75m + 8. What is the practical interpretation of the 8? What is the approximate weight of a baby at 3 months old?

Learn how a quadratic function approximates the weight of a baby during the first year of life. The formula w(m) = -0.042m^2 + 1.75m + 8 gives insight into the baby's weight at birth and its growth over time. This problem involves basic algebra and function substitution. Suitable for Grade 10 students.

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