24) Consider the following equation f(x) = x * sin(3x) whose roots we seek, answer the following question: with the initial bracket [2.5 , 3.5] how many bisections will it take to achieve an error tolerance of 10^-5?
a) 18 b) 16 c) 13 d) 17 e) 5

To determine the number of bisections needed to achieve a given error tolerance in the interval \([a, b]\) using the Bisection Method, we use the following formula:

\[
n \geq \frac{\log(b - a) - \log(\epsilon)}{\log(2)}
\]

where:
- \( n \) is the number of bisections,
- \( [a, b] = [2.5, 3.5] \) is the initial bracket,
- \( \epsilon = 10^{-5} \) is the error tolerance.

### Step 1: Calculate the width of the initial bracket
\[
b - a = 3.5 - 2.5 = 1.0
\]

### Step 2: Plug values into the formula
\[
n \geq \frac{\log(1.0) - \log(10^{-5})}{\log(2)}
\]

Simplify the logs:
\[
\log(1.0) = 0 \quad \text{and} \quad \log(10^{-5}) = -5
\]

Thus, the formula becomes:
\[
n \geq \frac{0 - (-5)}{\log(2)} = \frac{5}{\log(2)}
\]

Since \(\log(2) \approx 0.3010\), we get:
\[
n \geq \frac{5}{0.3010} \approx 16.61
\]

### Step 3: Round up
Since \(n\) must be an integer, round up to the next whole number. Thus, \(n = 17\).

### Conclusion:
The number of bisections needed is \( \boxed{17} \), so the correct answer is **d) 17**.

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Would you like further details on how bisection works or any other questions on this? 

Here are 5 related questions:
1. How does the Bisection Method guarantee convergence?
2. What is the formula for error reduction in the Bisection Method?
3. Can you apply the Bisection Method to discontinuous functions?
4. What are other root-finding methods that might be faster than bisection?
5. How can you determine whether the Bisection Method will work for a given function?

**Tip:** Always check the signs of \(f(a)\) and \(f(b)\) in the Bisection Method to ensure the function crosses the x-axis in the interval.

Discover how to calculate the number of bisections required to achieve an error tolerance of 10^-5 for the function f(x) = x * sin(3x) using the Bisection Method. This problem involves numerical methods and logarithms, suitable for advanced high school or university-level students.

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