The mean of the numbers 50, 40, 35, x + 10, x + 8, 12, 11, 8, 6 is 30. If median of the data is `n^2` - 1, then find the positive value of 3n

Answer

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

### 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:
\[ \text{Mean} = \frac{\text{Sum of all observations}}{\text{Number of observations}} \]

Here, the number of observations is 9. Therefore, 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
Now, let's calculate the sum of the constant numbers:
\[ 50 + 40 + 35 + 10 + 8 + 12 + 11 + 8 + 6 = 180 \]
So, we can rewrite the equation as:
\[ 30 = \frac{180 + 2x}{9} \]

### Step 3: Multiply Both Sides by 9
To eliminate the fraction, multiply both sides by 9:
\[ 270 = 180 + 2x \]

### Step 4: Solve for \( x \)
Now, isolate \( 2x \):
\[ 2x = 270 - 180 \]
\[ 2x = 90 \]
\[ x = 45 \]

### Step 5: Substitute \( x \) Back into the Observations
Now substitute \( x \) back into the observations:
\[ x + 10 = 55 \quad \text{and} \quad x + 8 = 53 \]
The updated list of numbers is:
\[ 50, 40, 35, 55, 53, 12, 11, 8, 6 \]

### Step 6: Arrange the Numbers in Ascending Order
Arranging the numbers in ascending order:
\[ 6, 8, 11, 12, 35, 40, 50, 53, 55 \]

### Step 7: Find the Median
The median is the middle value in an ordered list. Since there are 9 numbers, the median is the 5th number:
\[ \text{Median} = 35 \]

### Step 8: Set Up the Equation for Median
We know from the problem that the median is given as \( n^2 - 1 \):
\[ n^2 - 1 = 35 \]

### Step 9: Solve for \( n \)
Now, solve for \( n^2 \):
\[ n^2 = 35 + 1 \]
\[ n^2 = 36 \]
Taking the square root:
\[ n = 6 \quad (\text{positive value}) \]

### Step 10: Find \( 3n \)
Finally, calculate \( 3n \):
\[ 3n = 3 \times 6 = 18 \]

### Final Answer
The positive value of \( 3n \) is:
\[ \boxed{18} \]