The break even number of units is 0.6 and 15 which produce a maximum profit is 154
Total cost is the sum of the fixed cost and variable cost.
Let us assume that Q units are produced, hence:
Fixed cost = 30, variable cost = Q * (Q + 3) = Q² + 3Q
Total cost = Fixed cost + variable cost = 30 + (Q² + 3Q) = Q² + 3Q + 30
Given the demand function:
P + 2Q = 50
P = 50 - 2Q
Therefore the Revenue (R) = P * Q = (50 - 2Q) * (Q) = 50Q - 2Q²
The profit is the difference between the revenue and the cost, hence:
profit (p) = revenue - cost = (50Q - 2Q²) - (Q² + 3Q + 30) = -3Q² + 47Q - 30
p = -3Q² + 47Q - 30
At break even, the revenue is equal to the cost, hence no profit. At breakeven:
revenue = cost
50Q - 2Q² = Q² + 3Q + 30
3Q² - 47Q + 30 = 0
Q = 0.6 or 15
At breakeven the number of units produced are 0.6 and 15
The maximum profit is at dp/dQ = 0
p = -3Q² + 47Q - 30
dp/dQ = -6Q + 47 = 0
6Q = 47
Q = 7.83
p = -3(7.83)² + 47(7.83) - 30 = 154
Therefore the maximum profit is 154
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