A manager estimates the daily output of their robot model UOR402 at one of their factories to be given by: $$ Q=3 x^{3}+x^{2} y+y^{3} \text { (units), } $$ where $$ x $$ is the number of hours of skilled labor and $$ y $$ is the number of hours of unskilled labor. Currently, the factory is using 40 hours of skilled labor and 70 hours of unskilled labor. In order for the manager to maintain the current level of output of robot model UOR402, unskilled labor should be select an answer by approximately hundredth of an hour.

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To determine how much unskilled labor should be adjusted to maintain the current output of the robot model UOR402, we will follow these steps:

1. **Calculate the current output $$ Q $$** using the given values of $$ x $$ and $$ y $$.
2. **Differentiate $$ Q $$** with respect to $$ y $$ to find the marginal output with respect to unskilled labor.
3. **Set up the equation** to find the required change in $$ y $$ to maintain the same output when $$ x $$ is held constant.

### Step 1: Calculate the current output $$ Q $$

Given:
- $$ x = 40 $$
- $$ y = 70 $$

We can substitute these values into the output equation:

$$
Q = 3x^3 + x^2y + y^3
$$

### Step 2: Differentiate $$ Q $$ with respect to $$ y $$

The partial derivative of $$ Q $$ with respect to $$ y $$ is:

$$
\frac{\partial Q}{\partial y} = x^2 + 3y^2
$$

### Step 3: Set up the equation

To maintain the same output, we need to find the change in $$ y $$ (let's denote it as $$ \Delta y $$) such that:

$$
\Delta Q = \frac{\partial Q}{\partial y} \cdot \Delta y = 0
$$

This means we need to find $$ \Delta y $$ when $$ Q $$ remains constant.

Now, let's calculate $$ Q $$ and $$ \frac{\partial Q}{\partial y} $$ at the current values of $$ x $$ and $$ y $$.

Let's perform these calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$40^{2}+3\times 70^{2}$$
- step1: Multiply the terms:
  $$40^{2}+14700$$
- step2: Evaluate the power:
  $$1600+14700$$
- step3: Add the numbers:
  $$16300$$
Calculate or simplify the expression $$ 3*(40^3) + (40^2)*(70) + (70^3) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$3\times 40^{3}+40^{2}\times 70+70^{3}$$
- step1: Multiply the terms:
  $$192000+40^{2}\times 70+70^{3}$$
- step2: Multiply the terms:
  $$192000+112000+70^{3}$$
- step3: Add the numbers:
  $$304000+70^{3}$$
### Current Output Calculation

From the calculations, we found:

$$
Q = 3(40^3) + (40^2)(70) + (70^3) = 647000 \text{ units}
$$

### Marginal Output with Respect to Unskilled Labor

The partial derivative of $$ Q $$ with respect to $$ y $$ is:

$$
\frac{\partial Q}{\partial y} = 40^2 + 3(70^2) = 16300
$$

### Setting Up the Equation

To maintain the same output, we need:

$$
\Delta Q = \frac{\partial Q}{\partial y} \cdot \Delta y = 0
$$

Since we want to maintain the same output, we can express the change in output as:

$$
0 = 16300 \cdot \Delta y
$$

This means that any change in $$ y $$ will affect the output. However, we need to find the specific adjustment in $$ y $$ that keeps $$ Q $$ constant when $$ x $$ is held constant.

### Solving for $$ \Delta y $$

To maintain the output, we can rearrange the equation to find $$ \Delta y $$:

$$
\Delta y = 0
$$

This indicates that if we want to maintain the output with the current levels of skilled labor, no change in unskilled labor is necessary.

However, if the question implies a specific adjustment to maintain output under different conditions, we would need to know the new conditions to calculate $$ \Delta y $$.

### Conclusion

Under the current conditions of $$ x = 40 $$ and $$ y = 70 $$, the unskilled labor should remain at approximately **70 hours** to maintain the current output of 647000 units. If there are any specific changes in skilled labor or other conditions, please provide those details for further calculations.
To maintain the current output, unskilled labor should remain at approximately 70 hours.

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