Understanding the Geometric Mean
The geometric mean of a set of numbers is calculated by multiplying all the numbers together and then taking the nth root of the product, where n is the number of values. In this case, we are looking for the geometric mean of the values 40, 50, and x.
Setting Up the Equation
Given that the geometric mean of 40, 50, and x is 10, we can express this mathematically as:

• Geometric Mean = (40 * 50 * x)^(1/3) = 10

Solving the Equation
To solve for x, we first cube both sides to eliminate the cube root:

• (40 * 50 * x) = 10^3
• (40 * 50 * x) = 1000

Now, we can calculate the product of 40 and 50:

• 40 * 50 = 2000

Substituting this back into the equation gives:

• 2000 * x = 1000

Next, we isolate x by dividing both sides by 2000:

• x = 1000 / 2000
• x = 0.5

Conclusion
The value of x that satisfies the condition that the geometric mean of 40, 50, and x is 10 is:

• x = 0.5

This process illustrates how to manipulate equations involving the geometric mean to find an unknown variable.