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This problem involves related rates and a cubic relationship between revenue and the number of employees. Let's break it down step by step:

We are given the equation that relates revenue $ R $ (in thousands of USD) to the number of employees $ x $ (in hundreds):
\[
R^2 = 243x^3
\]
where:
- $ R $ is the revenue in thousands of USD,
- $ x $ is the number of employees in hundreds.

### Problem Details:
- The company currently has 2700 employees, so $ x = 27 $ (since $ x $ is in hundreds).
- The revenue is currently $2,187,000, so $ R = 2187 $ (since $ R $ is in thousands).
- The company wants to increase its revenue to $4,131,000 next year, meaning the desired future revenue is $ R = 4131 $.

We need to determine how many new employees need to be hired to achieve this goal.

### Step 1: Current Situation
Start with the equation $ R^2 = 243x^3 $ and plug in the current values:
\[
2187^2 = 243 \cdot 27^3
\]

Let's confirm this:
\[
2187^2 = 4782969, \quad 27^3 = 19683, \quad 243 \cdot 19683 = 4782969.
\]
So, the current equation holds true.

### Step 2: Desired Future Situation
Now, we want to find the number of employees required to achieve a revenue of $4,131,000. Use the same equation:
\[
4131^2 = 243x^3
\]

First, calculate $ 4131^2 $:
\[
4131^2 = 17064441
\]
Now, solve for $ x^3 $:
\[
x^3 = \frac{17064441}{243} = 70281
\]

Now, take the cube root of both sides:
\[
x = \sqrt[3]{70281} \approx 41.52
\]

### Step 3: Calculate the Number of New Employees
The current number of employees $ x = 27 $ (hundreds). To achieve the desired revenue, $ x $ needs to be about 41.52 (hundreds).

So, the number of new employees required is:
\[
41.52 - 27 = 14.52 \text{ hundred employees} = 1452 \text{ new employees}.
\]

### Conclusion:
The company should hire approximately **1452 new employees** to meet the revenue goal of $4,131,000 next year.

---

Would you like further clarification or details? Here are some related questions:

1. How would a change in the rate of hiring affect the revenue growth rate?
2. What if the company wanted to reach $5,000,000 in revenue instead?
3. How would the equation change if employee efficiency increased over time?
4. What if the company can only hire 1000 employees next year—how much revenue can they expect?
5. How does the revenue scale with respect to changes in employee numbers based on the cubic relationship?

**Tip:** When dealing with cubic relationships, small changes in the input (like employees) can lead to much larger changes in the output (like revenue), as seen in this problem.

A technical support contracting firm hires people to work from home using their proprietary support scripting system. The company currently has 2700 employees, and the company wants to increase their revenue from $2,187,000 this year to $4,131,000 next year. How many new employees should be hired to achieve this?

Solve a related rates problem involving a cubic relationship between the number of employees and company revenue. Learn how to determine the number of new hires required to increase revenue from $2,187,000 to $4,131,000, based on a formula relating revenue and employees.

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