A stretched string is taken and stretched such that elongated by 1% then the fundamental frequency decreased by `x xx10^(-1) %` what is the value of x ?

Answer

To solve the problem step by step, we will analyze the relationship between the length of the string and its fundamental frequency.

### Step 1: Understand the relationship between frequency and length
The fundamental frequency \( f \) of a stretched string is given by the formula:
\[ f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \]
where:
- \( L \) is the length of the string,
- \( T \) is the tension in the string,
- \( \mu \) is the mass per unit length.

From this formula, we can see that the frequency is inversely proportional to the length of the string. Thus, if the length increases, the frequency decreases.

### Step 2: Determine the new length of the string
The string is elongated by 1%. If the original length of the string is \( L \), the new length \( L_2 \) can be expressed as:
\[ L_2 = L + 0.01L = 1.01L \]

### Step 3: Relate the initial and new frequencies
Let the initial frequency be \( f_1 \) and the new frequency after stretching be \( f_2 \). Since frequency is inversely proportional to length, we can write:
\[ f_1 L_1 = f_2 L_2 \]
Substituting \( L_1 = L \) and \( L_2 = 1.01L \):
\[ f_1 L = f_2 (1.01L) \]
Cancelling \( L \) from both sides gives:
\[ f_1 = f_2 \cdot 1.01 \]
Thus, we can express \( f_2 \) in terms of \( f_1 \):
\[ f_2 = \frac{f_1}{1.01} \]

### Step 4: Calculate the decrease in frequency
The decrease in frequency \( \Delta f \) can be calculated as:
\[ \Delta f = f_1 - f_2 = f_1 - \frac{f_1}{1.01} \]
Factoring out \( f_1 \):
\[ \Delta f = f_1 \left( 1 - \frac{1}{1.01} \right) \]
Calculating the term in parentheses:
\[ 1 - \frac{1}{1.01} = \frac{1.01 - 1}{1.01} = \frac{0.01}{1.01} \]
Thus, we have:
\[ \Delta f = f_1 \cdot \frac{0.01}{1.01} \]

### Step 5: Express the decrease in percentage
To express the decrease in frequency as a percentage of \( f_1 \):
\[ \text{Percentage decrease} = \frac{\Delta f}{f_1} \times 100 = \frac{0.01}{1.01} \times 100 \]
Calculating this gives:
\[ \text{Percentage decrease} \approx 0.9901\% \]

### Step 6: Relate to the given format
We are given that the decrease in frequency can be expressed as \( x \times 10^{-1} \% \). Therefore, we can write:
\[ 0.9901\% \approx x \times 10^{-1} \% \]
This implies:
\[ x \approx 0.9901 \times 10 = 9.901 \]
Rounding to one decimal place gives:
\[ x \approx 0.1 \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{0.1} \]