The fixed cost of a new product is Rs. 35,000 and the variable cost per unit is Rs. 500. If the demand function is p = 5000 - 100x, where x is the item demanded. Find the break-even point.
Answers
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The break-even point occurs when 33 units are sold.
To find the break-even point, we need to determine the quantity of units that need to be sold to cover the total cost of producing those units.
Let's begin by calculating the total cost (TC) for a given quantity (x) of units:
TC = Fixed Cost + Variable Cost
= 35,000 + 500x
The revenue (R) generated from selling x units is given by the demand function:
R = px
= (5000 - 100x)x
= 5000x - 100x^2
The break-even point is reached when all costs and revenues are equal.
R = TC
5000x - 100x^2 = 35,000 + 500x
Simplifying the equation by bringing all the terms to one side:
100x^2 - 4500x + 35,000 = 0
Solving for x using the quadratic formula:
x=(-b sqrt(b2 - 4ac)) / 2a
= (-(-4500) ± sqrt((-4500)^2 - 4(100)(35000))) / 2(100)
= (4500 ± sqrt(4500^2 - 410035000)) / 200
= (4500 ± 2100) / 200
The positive root of the equation is:
x = 33
Therefore, the break-even point occurs when 33 units are sold.
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