Answer:
Step-by-step explanation:
Let x represent the number of tubs of ice cream that was produced.
The unit cost, in dollars, to produce tubs of ice cream is $18 and the fixed cost is $11610. This means that the total cost of producing x tubs of ice cream would be
C(x) = 18x + 11610
The price-demand function, in dollars per tub, is p(x)=374-2x.
The revenue function is product of the output by the price function
R(x) = x × p(x) = xp(x)
R(x) = x(374 - 2x) = 374x - 2x²
The profit function P(x) = R(x) - C(x)
Therefore,
P(x) = 374x - 2x² - (18x + 11610)
P(x) = 374x - 2x² - 18x - 11610
P(x) = - 2x² + 374x - 18x - 11610
P(x) = - 2x² + 356x - 11610
At the break even point,
Revenue = total cost.
Therefore,
374x - 2x² = 18x + 11610
2x² + 18x - 374x + 11610 = 0
2x² - 356x + 11610 = 0
Dividing through by 2, it becomes
x² - 178x + 5805 = 0
Applying the general formula for quadratic equations,
x = [- b ±√(b² - 4ac)]/2a
x = [- - 178 ±√(-178² - 4 × 1 × 5805)]/2 × 1
x = [178 ±√(31684 - 23220)]/2
x = [178 ±92]/2
x = (178 + 92)/2 or (178 - 92)/2
x = 135 or x = 43
Therefore, the quantity for the smallest break-even point is 43.
