The demand function for a commodity is Q = 6000 - 30P, with P the price and Q the number of units. Fixed costs are R 72 000 and the variable cost is R 60 per additional unit produced. Dtermine the price at which profit is a maximum and calculate the maximum profit.

The demand function for a commodity is Q = 6000 - 30P, with P the price and Q the number of units. Fixed costs are R 72 000 and the variable cost is R 60 per additional unit produced. Dtermine the price at which profit is a maximum and calculate the maximum profit.

Question
Ratios, rates, proportions
asked 2021-01-04
The demand function for a commodity is Q = 6000 - 30P, with P the price and Q the number of units. Fixed costs are R 72 000 and the variable cost is R 60 per additional unit produced. Dtermine the price at which profit is a maximum and calculate the maximum profit.

Answers (1)

2021-01-05
Revenue earned will be = Demand * Price Revenue = (6000 - 30P) * PZSK
Total Cost = Fixed Cost + Variable Cost
Total Cost =\(\displaystyle{72000}+{60}\cdot{\left({6000}-{30}{P}\right)}\)
Profit = Revenue - Total Cost
Profit = \(\displaystyle{\left({6000}-{30}{P}\right)}\cdot{P}-{\left({72000}+{60}\cdot{\left({6000}-{30}{P}\right)}\right)}\)
Profit = - \(\displaystyle{30}{P}^{{2}}+{7800}{P}-{432000}\)
Now we have a quadratic equation for Profit. Now its a simple case of maximum value in quadratic equation.
Maximum Value of quadratic equation is at x = \(\displaystyle-\frac{{b}}{{2}}{a}\)
At P = \(\displaystyle-\frac{{7800}}{{{2}\cdot-{30}}}\), we will get max profit
i.e. Price at which max profit, P = R 130
Maximum Profit = R 75000
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