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The problem provides the function for the median weight \( w(t) \) of a boy whose age is between 0 and 36 months, where \( t \) is the age in months and \( w(t) \) is the weight in pounds:

\[
w(t) = 8.46 + 1.09t - 0.0009t^2 + 0.000843t^3
\]

We are asked to solve the following:

### (a) Rate of change of weight with respect to time (which is already given as \( w'(t) \)):
\[
w'(t) = 1.09 - 0.0018t + 0.002529t^2
\]

### (b) Weight of the baby at age 10 months:
We will substitute \( t = 10 \) into the original function \( w(t) \).

Let me calculate the weight at 10 months.The approximate weight of the baby at 10 months is \( 20.11 \) pounds.

Would you like more details on any part of the solution, or have any questions?

Here are five related questions for further practice:
1. What is the rate of change of the baby's weight at 10 months?
2. How would the baby's weight change if we extended the model to 40 months?
3. Calculate the weight of the baby at age 20 months using the same function.
4. What is the rate of change of the baby's weight at birth (i.e., at \( t = 0 \))?
5. How does the second derivative of the function \( w(t) \) provide insight into the acceleration of weight gain?

**Tip:** The derivative of a function gives the rate of change, and higher-order derivatives provide insights into the behavior of this rate over time.

The median weight of a boy whose age is between 0 and 36 months can be approximated by the function w(t) = 8.46 + 1.09t - 0.0009t^2 + 0.000843t^3, where t is measured in months and w is measured in pounds. Find the following for a boy with median weight: a) The rate of change of weight with respect to time. b) The weight of the baby at age 10 months.

Learn how to calculate the median weight of a baby whose age is between 0 and 36 months using the polynomial function w(t) = 8.46 + 1.09t - 0.0009t^2 + 0.000843t^3. This example also covers the rate of change in weight over time using basic differentiation rules. Suitable for high school or early college students.

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