What is the geometric mean of 10, 40 and 60 (appox)?

Options

A. 10
B. 28
C. 29.6
D. 70

Explanation

To find the geometric mean of the numbers 10, 40, and 60, we follow these steps:

### Step 1: Understand the formula for geometric mean
The geometric mean (GM) of n numbers \( x_1, x_2, x_3, \ldots, x_n \) is given by the formula:
\[ GM = \sqrt[n]{x_1 \times x_2 \times x_3 \times \ldots \times x_n} \]
In this case, we have three numbers, so \( n = 3 \).

### Step 2: Identify the numbers
The numbers we need to find the geometric mean for are:
- \( x_1 = 10 \)
- \( x_2 = 40 \)
- \( x_3 = 60 \)

### Step 3: Calculate the product of the numbers
Now, we will calculate the product of these three numbers:
\[ x_1 \times x_2 \times x_3 = 10 \times 40 \times 60 \]

Calculating this step-by-step:
1. First, calculate \( 10 \times 40 = 400 \).
2. Then, multiply \( 400 \times 60 = 24000 \).

So, the product is:
\[ 10 \times 40 \times 60 = 24000 \]

### Step 4: Calculate the geometric mean
Now, we will use the product to find the geometric mean:
\[ GM = \sqrt[3]{24000} \]

### Step 5: Simplify the cube root
To simplify \( \sqrt[3]{24000} \), we can factor 24000:
\[ 24000 = 24 \times 1000 = 24 \times 10^3 \]
Next, we can break down 24:
\[ 24 = 2^3 \times 3 \]
Thus,
\[ 24000 = 2^3 \times 3 \times 10^3 = 2^3 \times 3 \times (2 \times 5)^3 = 2^3 \times 3 \times 2^3 \times 5^3 = 2^6 \times 3 \times 5^3 \]

Now, we can find the cube root:
\[ \sqrt[3]{24000} = \sqrt[3]{2^6} \times \sqrt[3]{3} \times \sqrt[3]{5^3} = 2^2 \times \sqrt[3]{3} \times 5 = 4 \times \sqrt[3]{3} \times 5 \]

Calculating \( 4 \times 5 = 20 \), we have:
\[ GM \approx 20 \times \sqrt[3]{3} \]

Using a calculator, \( \sqrt[3]{3} \approx 1.442 \), so:
\[ GM \approx 20 \times 1.442 \approx 28.84 \]

### Final Result
The approximate geometric mean of 10, 40, and 60 is:
\[ \text{GM} \approx 28.84 \]