A company has `MR = 30 x + 15 x^(2)` and `MC = 64 -1 6x + (3)/(2)x^(2)`. Find out the profit function and the output `x gt 0` when there is no profit if the fixed cost is zero.

Answer

To solve the problem step by step, we will find the profit function and determine the output when there is no profit, given that the fixed cost is zero.

### Step 1: Identify the Marginal Revenue (MR) and Marginal Cost (MC)
We are given:
- Marginal Revenue (MR) = \( 30x + 15x^2 \)
- Marginal Cost (MC) = \( 64 - 16x + \frac{3}{2}x^2 \)

### Step 2: Find the Revenue Function (R)
To find the revenue function, we integrate the marginal revenue function with respect to \( x \):
\[ R(x) = \int (30x + 15x^2) \, dx \]
Calculating the integral:
\[ R(x) = 15x^2 + 5x^3 + C \]
where \( C \) is the constant of integration. Since fixed costs are zero, we can set \( C = 0 \):
\[ R(x) = 15x^2 + 5x^3 \]

### Step 3: Find the Cost Function (C)
Next, we integrate the marginal cost function to find the cost function:
\[ C(x) = \int (64 - 16x + \frac{3}{2}x^2) \, dx \]
Calculating the integral:
\[ C(x) = 64x - 8x^2 + \frac{1}{2}x^3 + D \]
Again, since fixed costs are zero, we can set \( D = 0 \):
\[ C(x) = 64x - 8x^2 + \frac{1}{2}x^3 \]

### Step 4: Find the Profit Function (P)
The profit function \( P(x) \) is defined as the revenue function minus the cost function:
\[ P(x) = R(x) - C(x) \]
Substituting the expressions for \( R(x) \) and \( C(x) \):
\[ P(x) = (15x^2 + 5x^3) - (64x - 8x^2 + \frac{1}{2}x^3) \]
Simplifying this:
\[ P(x) = 15x^2 + 5x^3 - 64x + 8x^2 - \frac{1}{2}x^3 \]
Combining like terms:
\[ P(x) = (5 - \frac{1}{2})x^3 + (15 + 8)x^2 - 64x \]
\[ P(x) = \frac{9}{2}x^3 + 23x^2 - 64x \]

### Step 5: Set the Profit Function to Zero
To find the output when there is no profit, we set \( P(x) = 0 \):
\[ \frac{9}{2}x^3 + 23x^2 - 64x = 0 \]
Factoring out \( x \):
\[ x\left(\frac{9}{2}x^2 + 23x - 64\right) = 0 \]
This gives us one solution \( x = 0 \) (which we discard since \( x > 0 \)) and we need to solve the quadratic equation:
\[ \frac{9}{2}x^2 + 23x - 64 = 0 \]

### Step 6: Solve the Quadratic Equation
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = \frac{9}{2}, b = 23, c = -64 \):
\[ b^2 - 4ac = 23^2 - 4 \cdot \frac{9}{2} \cdot (-64) \]
Calculating:
\[ = 529 + 1152 = 1681 \]
Now substituting into the formula:
\[ x = \frac{-23 \pm \sqrt{1681}}{2 \cdot \frac{9}{2}} = \frac{-23 \pm 41}{9} \]
Calculating the two possible values:
1. \( x_1 = \frac{18}{9} = 2 \)
2. \( x_2 = \frac{-64}{9} \) (discarded since it's negative)

### Conclusion
The output when there is no profit is \( x = 2 \).

### Final Answer
- Profit Function: \( P(x) = \frac{9}{2}x^3 + 23x^2 - 64x \)
- Output when profit equals zero: \( x = 2 \)