Let's break this down and solve the problem step by step.

Given Information:

1. The relationship between revenue $R$ (in thousands of USD) and the number of employees $x$ (in hundreds) is given by: $R^2 = 270x^3$
2. The company currently has 3000 employees, which means $x = 30$ (since $x$ is the number of employees in hundreds).
3. The current revenue is $2,700,000, which means $R = 2700$ (since $R$ is in thousands).
4. The target revenue is $4,860,000, which means $R = 4860$.

We need to find how many new employees should be hired, i.e., find the new value of $x$.

Step 1: Solve for $x$ at the current revenue.

From the equation: $R^2 = 270x^3$ Substitute $R = 2700$ and solve for $x$: $(2700)^2 = 270x^3$ $7290000 = 270x^3$ $x^3 = \frac{7290000}{270} = 27000$ $x = \sqrt[3]{27000} = 30$ This confirms that with 3000 employees (i.e., $x = 30$), the revenue is indeed $2,700,000.

Step 2: Solve for $x$ at the target revenue.

Now, substitute $R = 4860$ and solve for $x$: $(4860)^2 = 270x^3$ $23619600 = 270x^3$ $x^3 = \frac{23619600}{270} = 87480$ $x = \sqrt[3]{87480} \approx 44.59$

Step 3: Determine how many employees to hire.

The company needs $x = 44.59$, which corresponds to approximately 4459 employees. Currently, they have 3000 employees, so the number of new employees to hire is: $4459 - 3000 = 1459$

Thus, the company should hire 1459 new employees to reach their target revenue.

Would you like further clarification or a deeper breakdown of any step?

Here are five related questions for further practice:

1. What if the target revenue was $6,000,000? How many new employees would be needed?
2. If the revenue-to-employees relationship changes to $R^2 = 350x^3$, how does this affect the number of employees needed for the same revenue?
3. How many employees would the company need if they wanted to double their current revenue?
4. What would the revenue be if the company hired an additional 2000 employees?
5. How would the equation change if the revenue grows at a rate proportional to $x^4$ instead of $x^3$?

Tip: When dealing with related rates or proportional growth problems, always check if the units are consistent to avoid errors in calculations.