The number of units sold, N, of a product follows the realtion N= 120-C, where $C is the selling price per unit. The cost to setup the manufacturing facility is $150 and the cost per unit is $5. If all units are sold, what should be the least selling price, in dollars, per unit to have a profit of $400?

Answer

To find the least selling price per unit \( C \) that results in a profit of $400, we can follow these steps:

### Step 1: Define the variables and equations
- The number of units sold \( N \) is given by the equation:
\[ N = 120 - C \]
- The total cost \( TC \) includes the setup cost and the cost per unit:
\[ TC = 150 + 5N = 150 + 5(120 - C) \]
- The total revenue \( TR \) from selling the units is:
\[ TR = C \cdot N = C(120 - C) \]
- The profit \( P \) is defined as:
\[ P = TR - TC \]

### Step 2: Set up the profit equation
We want the profit to be $400:
\[ P = TR - TC = 400 \]
Substituting the equations for \( TR \) and \( TC \):
\[ C(120 - C) - (150 + 5(120 - C)) = 400 \]

### Step 3: Simplify the equation
Expanding the equation:
\[ C(120 - C) - 150 - 600 + 5C = 400 \]
This simplifies to:
\[ 120C - C^2 - 750 + 5C = 400 \]
Combining like terms:
\[ - C^2 + 125C - 750 = 400 \]
Rearranging gives:
\[ - C^2 + 125C - 1150 = 0 \]
Multiplying through by -1:
\[ C^2 - 125C + 1150 = 0 \]

### Step 4: Solve the quadratic equation
Using the quadratic formula \( C = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = -125, c = 1150 \):
\[ C = \frac{125 \pm \sqrt{(-125)^2 - 4 \cdot 1 \cdot 1150}}{2 \cdot 1} \]
Calculating the discriminant:
\[ C = \frac{125 \pm \sqrt{15625 - 4600}}{2} \]
\[ C = \frac{125 \pm \sqrt{11025}}{2} \]
\[ C = \frac{125 \pm 105}{2} \]
Calculating the two potential solutions:
1. \( C = \frac{230}{2} = 115 \)
2. \( C = \frac{20}{2} = 10 \)

### Step 5: Determine the least selling price
The two potential selling prices are $115 and $10. Since we want the least selling price to achieve a profit of $400:
\[ \text{Least selling price } C = 10 \]

Thus, the least selling price per unit to achieve a profit of $400 is **$10**.