The mean of the numbers 50, 40, 35, x + 10, x + 8, 12, 11, 8, 6 is 30. It median of the data is `n^2+ 10,` then the positive value of n is_____

Answer

To solve the problem step by step, we will first find the value of \(x\) using the information about the mean, and then we will find the median to determine the positive value of \(n\).

### Step 1: Calculate the Mean
The mean of the numbers is given as 30. The numbers are:
\[ 50, 40, 35, x + 10, x + 8, 12, 11, 8, 6 \]

The formula for the mean is:
\[ \text{Mean} = \frac{\text{Sum of all observations}}{\text{Number of observations}} \]

The number of observations is 9. Thus, we can set up the equation:
\[ 30 = \frac{50 + 40 + 35 + (x + 10) + (x + 8) + 12 + 11 + 8 + 6}{9} \]

### Step 2: Simplify the Equation
Calculating the sum of the known numbers:
\[ 50 + 40 + 35 + 10 + 8 + 12 + 11 + 8 + 6 = 50 + 40 + 35 + 10 + 8 + 12 + 11 + 8 + 6 = 270 + 2x \]

So we have:
\[ 30 = \frac{270 + 2x}{9} \]

### Step 3: Cross Multiply
Cross-multiplying gives:
\[ 30 \times 9 = 270 + 2x \]
\[ 270 = 270 + 2x \]

### Step 4: Solve for \(x\)
Subtracting 270 from both sides:
\[ 0 = 2x \]
\[ x = 0 \]

### Step 5: Substitute \(x\) Back into the Observations
Now substituting \(x = 0\) back into the observations:
\[ 50, 40, 35, 0 + 10 = 10, 0 + 8 = 8, 12, 11, 8, 6 \]
This gives us the numbers:
\[ 50, 40, 35, 10, 8, 12, 11, 8, 6 \]

### Step 6: Arrange the Numbers in Increasing Order
Arranging the numbers in increasing order:
\[ 6, 8, 8, 10, 11, 12, 35, 40, 50 \]

### Step 7: Find the Median
The median is the middle value. Since there are 9 numbers (odd), the median is the 5th number:
\[ \text{Median} = 11 \]

### Step 8: Set Up the Equation for Median
According to the problem, the median is also given as \(n^2 + 10\):
\[ n^2 + 10 = 11 \]

### Step 9: Solve for \(n\)
Subtracting 10 from both sides:
\[ n^2 = 1 \]
Taking the square root gives:
\[ n = \pm 1 \]
Since we need the positive value:
\[ n = 1 \]

### Final Answer
The positive value of \(n\) is:
\[ \boxed{1} \]