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For part (e), the problem asks to calculate the rate of change of the tumor volume at 240 months. This refers to finding the derivative of the volume function \( V(t) \) with respect to time \( t \), and then evaluating it at \( t = 240 \).

The volume function given is:

\[
V(t) = 1100 \left[ 1023 e^{-0.02415t} + 1 \right]^{-4}
\]

### Steps:

1. **Differentiate \( V(t) \) with respect to \( t \):**

   We will use the chain rule to differentiate \( V(t) \). Notice that the function is of the form:

   \[
   V(t) = 1100 \left( f(t) \right)^{-4}
   \]
   
   where \( f(t) = 1023 e^{-0.02415t} + 1 \). 

   The derivative of \( V(t) \) can be computed using the chain rule:
   
   \[
   \frac{dV}{dt} = 1100 \cdot (-4) \cdot \left( f(t) \right)^{-5} \cdot f'(t)
   \]
   
   Now we need to compute \( f'(t) \).

2. **Differentiate \( f(t) = 1023 e^{-0.02415t} + 1 \):**
   
   The derivative of \( f(t) \) is:

   \[
   f'(t) = 1023 \cdot (-0.02415) e^{-0.02415t}
   \]

   Simplifying:

   \[
   f'(t) = -24.71 e^{-0.02415t}
   \]

3. **Substitute \( f(t) \) and \( f'(t) \) back into the derivative of \( V(t) \):**

   Now, substitute \( f(t) = 1023 e^{-0.02415t} + 1 \) and \( f'(t) = -24.71 e^{-0.02415t} \) into the equation for \( \frac{dV}{dt} \):

   \[
   \frac{dV}{dt} = 1100 \cdot (-4) \cdot \left( 1023 e^{-0.02415t} + 1 \right)^{-5} \cdot (-24.71 e^{-0.02415t})
   \]

   Simplifying the constants:

   \[
   \frac{dV}{dt} = 4 \cdot 1100 \cdot 24.71 e^{-0.02415t} \cdot \left( 1023 e^{-0.02415t} + 1 \right)^{-5}
   \]

4. **Evaluate the derivative at \( t = 240 \):**

   Substitute \( t = 240 \) into the expression:

   - First, compute \( f(240) = 1023 e^{-0.02415 \cdot 240} + 1 \).
   - Then, calculate \( e^{-0.02415 \cdot 240} \).
   - Finally, plug everything into the derivative expression.

Let me calculate this step-by-step for you.The rate of change of the tumor volume at 240 months is approximately \( 0.282 \, \text{cm}^3/\text{month} \). This means that at 240 months, the tumor is growing at a rate of about 0.282 cubic centimeters per month.

Would you like more details on any specific step, or do you have any questions?

Here are 5 related questions you might find helpful:

1. What is the biological significance of the tumor growth rate at 240 months?
2. How would the rate of change differ at an earlier time, like 100 months?
3. Can we determine when the tumor's growth rate was at its peak?
4. How does the exponential decay factor \( e^{-0.02415t} \) affect long-term tumor growth?
5. What would the tumor volume be after 500 months?

**Tip**: Always check the units of the derivative when calculating rates of change, to ensure consistency in your interpretation of the results.

This page provides a detailed step-by-step solution for calculating the rate of change of tumor volume after 240 months using the formula V(t) = 1100 [1023 e^{-0.02415t} + 1]^{-4}. Explore key calculus concepts like differentiation and the chain rule.

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