To determine $x^2$, the square of the cost of labor per hour, we'll follow these steps:

Step 1: Define the variables and equation

• Let $x$ represent the cost of labor per hour.
• The technician worked for 2.5 hours, so the total labor cost would be $2.5x$.
• The parts cost $100.
• The total cost was $237.50.

We can set up the following equation to represent the total cost:

$2.5x + 100 = 237.50$

Step 2: Solve for $x$

Subtract $100 from both sides to isolate the labor cost:

$2.5x = 237.50 - 100$

$2.5x = 137.50$

Now, divide both sides by 2.5 to solve for $x$:

$x = \frac{137.50}{2.5} = 55$

Step 3: Find $x^2$

Now that we have $x = 55$, the square of the cost of labor per hour is:

$x^2 = 55^2 = 3025$

So, $x^2 = 3025$, meaning the square of the labor cost per hour is $3025.

Would you like more details on any of these steps, or do you have any questions?

Here are 5 related questions for further practice:

1. If the total cost was $300, what would be the new value of $x^2$?
2. What would the square of the labor cost per hour be if the technician worked for 3 hours and the total cost was $250?
3. If the parts cost increased to $150, how would that affect $x^2$ if the total remained $237.50?
4. If $x^2$ was calculated to be $3600, what would be the cost per hour?
5. How would the total cost change if the technician charged $50 per hour and worked for 2.5 hours, and $x^2$ was used in the calculation?

Tip: When dealing with squares in equations, remember that $x^2$ represents the area of a square with side length $x$, and it's important to solve for $x$ first before squaring it.