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The uploaded document contains a problem related to the altitude of a rocket using a given velocity function. The function provided is:

\[
v(t) = u \ln \left(\frac{m}{m - qt}\right) - gt
\]

Where:
- \( u = 1800 \, \text{m/s} \) is the velocity relative to the rocket,
- \( m = 160000 \, \text{kg} \) is the initial mass of the rocket,
- \( q = 2500 \, \text{kg/s} \) is the rate of fuel consumption,
- \( g = 9.8 \, \text{m/s}^2 \) is the acceleration due to gravity,
- \( t \) is time in seconds.

### Key tasks:
1. **4th-order derivative**: You need to find the 4th derivative of the velocity function and compute the maximum of its modulus.
2. **Minimum intervals for accuracy**: Determine the minimum number of intervals \( n \) for Simpson's rule to ensure accuracy up to two decimal places.
3. **Simpson’s Rule setup**: Calculate the altitude of the rocket after 30 seconds by applying the 1/3 Simpson’s rule to approximate the integral.
4. **Tabulate results**: Present the data points used for the numerical method based on the computed \( n \).
5. **Matlab confirmation**: Verify the solution using Matlab.

Would you like help with the derivation, the numerical setup for Simpson's rule, or further guidance on any specific part?

### Follow-up questions:
1. How do we differentiate the given function multiple times to find the 4th derivative?
2. How is Simpson’s rule applied in this case, and why is it effective for this problem?
3. What factors influence the number of intervals \( n \) in Simpson's method?
4. How do we estimate the error in Simpson’s rule based on the derivative maximum?
5. What are the advantages of verifying the result using Matlab?

### Tip:
Simpson's rule is very efficient for smooth functions, as it approximates the integral using quadratic polynomials between intervals.

The velocity of a rocket is given by the function: v(t) = u ln(m / (m - qt)) - gt, where u = 1800 m/s, m = 160000 kg, q = 2500 kg/s, and g = 9.8 m/s². Use Simpson's 1/3 rule to calculate the rocket's altitude after 30 seconds. Key tasks include finding the 4th-order derivative, calculating the minimum number of intervals n, and verifying the result using Matlab.

Calculate the altitude of a rocket using the velocity function v(t) = u ln(m / (m - qt)) - gt, applying Simpson's 1/3 rule. This problem involves finding the 4th-order derivative and determining the minimum number of intervals for accurate approximation. Suitable for advanced university students.

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