A guitar string is 90 cm long and has a fundamental frequency of 124 Hz. Where should it be pressed to produce a fundamatal frequecy of 186 Hz?

Explanation

To solve the problem, we need to find the length at which the guitar string should be pressed to produce a fundamental frequency of 186 Hz, given that the original length of the string is 90 cm and its fundamental frequency is 124 Hz.

### Step-by-Step Solution:

1. **Understand the relationship between frequency and length**: The fundamental frequency (f) of a vibrating string is inversely proportional to its length (L). This can be expressed as:
\[ f \propto \frac{1}{L} \]
Therefore, we can write the relationship between two frequencies and their corresponding lengths as:
\[ \frac{f_1}{f_2} = \frac{L_2}{L_1} \]
where \( f_1 \) is the initial frequency, \( f_2 \) is the new frequency, \( L_1 \) is the initial length, and \( L_2 \) is the new length.

2. **Substitute the known values**: We know:
- \( f_1 = 124 \, \text{Hz} \)
- \( f_2 = 186 \, \text{Hz} \)
- \( L_1 = 90 \, \text{cm} \)

Plugging these values into the equation gives:
\[ \frac{124}{186} = \frac{L_2}{90} \]

3. **Cross-multiply to solve for \( L_2 \)**:
\[ 124 \cdot 90 = 186 \cdot L_2 \]
\[ 11160 = 186 \cdot L_2 \]

4. **Isolate \( L_2 \)**:
\[ L_2 = \frac{11160}{186} \]

5. **Calculate \( L_2 \)**:
\[ L_2 = 60 \, \text{cm} \]

6. **Conclusion**: To produce a fundamental frequency of 186 Hz, the string should be pressed at a length of 60 cm.

### Final Answer:
The guitar string should be pressed at 60 cm to produce a fundamental frequency of 186 Hz.