After experimentation, a certain manufacturer determined that if x units of a particular commodity are produced per week, the marginal cost is given by 0.3x -11, where the production cost is in dollars. If the selling price of the commodity is fixed at $19 per unit, and the fixed cost is $100 per week, find the maximum profit that can be obtained.
for x units, the profit p(x) is
p(x) = 19x-100-c(x)
where c(x) is the cost of the units.
now,
dc/dx = .3x-11, so c(x) = 0.15x^2-11x (why no +C?)
p(x) = 19x-100-(0.15x^2-11x)
= -0.15x^2 + 30x - 100
now just find the vertex of that parabola for maximum profit.
To find the maximum profit, we need to determine the optimal number of units to produce per week.
The profit can be calculated using the following formula:
Profit = Revenue - Total Cost
The revenue is given by the selling price multiplied by the number of units sold:
Revenue = Selling Price * Number of Units
In this case, the selling price is $19 per unit.
The total cost consists of two components: the fixed cost and the marginal cost multiplied by the number of units produced:
Total Cost = Fixed Cost + Marginal Cost * Number of Units
The fixed cost is given as $100 per week.
The marginal cost is given by 0.3x - 11, where x is the number of units produced.
Combining the revenue and total cost equations, we can now express the profit as a function of x:
Profit = (Selling Price * Number of Units) - (Fixed Cost + Marginal Cost * Number of Units)
Profit = (19x) - (100 + (0.3x - 11)x)
Profit = 19x - 100 - 0.3x^2 + 11x
To find the maximum profit, we need to find the value of x that maximizes this profit function. This can be done by taking the derivative of the profit function with respect to x, setting it equal to zero, and solving for x.
First, let's differentiate the profit function:
d(Profit)/dx = 19 - 0.6x + 11
Next, set the derivative equal to zero and solve for x:
19 - 0.6x + 11 = 0
-0.6x = -30
x = 50
Now that we have found the value of x that maximizes the profit, we can substitute it back into the profit function to find the maximum profit:
Profit = 19(50) - 100 - 0.3(50)^2 + 11(50)
Profit = 950 - 100 - 0.3(2500) + 550
Profit = 950 - 100 - 750 + 550
Profit = 650
Therefore, the maximum profit that can be obtained is $650 per week.