The fixed cost of a new product is Rs. 30,000 and the variable cost per unit is Rs. 800. If the demand function is `x=45-(p)/(100)`, find break-even values.

Answer

To find the break-even values for the given problem, we will follow these steps:

### Step 1: Define Fixed and Variable Costs
The fixed cost (FC) is given as Rs. 30,000 and the variable cost (VC) per unit is Rs. 800.

### Step 2: Write the Total Cost Function
The total cost (C) can be expressed as:
\[ C(x) = \text{Fixed Cost} + \text{Variable Cost} \times \text{Number of Units} \]
So,
\[ C(x) = 30000 + 800x \]

### Step 3: Define the Demand Function
The demand function is given as:
\[ x = 45 - \frac{p}{100} \]
We can rearrange this to express price (p) in terms of quantity (x):
\[ \frac{p}{100} = 45 - x \]
Multiplying both sides by 100 gives:
\[ p = 4500 - 100x \]

### Step 4: Write the Revenue Function
Revenue (R) is calculated as the product of price and quantity:
\[ R(x) = x \cdot p \]
Substituting the expression for p:
\[ R(x) = x(4500 - 100x) \]
Expanding this gives:
\[ R(x) = 4500x - 100x^2 \]

### Step 5: Set Up the Profit Function
Profit (P) is defined as Revenue minus Total Cost:
\[ P(x) = R(x) - C(x) \]
Substituting the expressions for R(x) and C(x):
\[ P(x) = (4500x - 100x^2) - (30000 + 800x) \]
Simplifying this gives:
\[ P(x) = 4500x - 100x^2 - 30000 - 800x \]
Combining like terms:
\[ P(x) = -100x^2 + 3700x - 30000 \]

### Step 6: Find Break-Even Point
At the break-even point, profit is zero:
\[ P(x) = 0 \]
So we set the profit function to zero:
\[ -100x^2 + 3700x - 30000 = 0 \]
To simplify, we can divide the entire equation by -100:
\[ x^2 - 37x + 300 = 0 \]

### Step 7: Solve the Quadratic Equation
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1, b = -37, c = 300 \).
Calculating the discriminant:
\[ b^2 - 4ac = (-37)^2 - 4 \cdot 1 \cdot 300 = 1369 - 1200 = 169 \]
Now substituting into the quadratic formula:
\[ x = \frac{37 \pm \sqrt{169}}{2 \cdot 1} \]
\[ x = \frac{37 \pm 13}{2} \]
Calculating the two possible values:
1. \( x = \frac{50}{2} = 25 \)
2. \( x = \frac{24}{2} = 12 \)

### Conclusion
The break-even values are \( x = 12 \) and \( x = 25 \).

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