Final answer:
a) The quantity that will maximize profit is 49.5 units. b) The selling price at the optimal quantity is -$639.5 per unit. c) The maximum profit is -$4851.
Step-by-step explanation:
a) Finding the quantity that will maximize profit:
To find the quantity that will maximize profit, we need to take the derivative of the profit function and set it equal to zero.
Profit function = revenue - cost
Profit function = (price x quantity) - (average cost x quantity)
Profit function = (400 - 21x)x - (200 + 2x)x
Profit function = 400x - 21x^2 - 200x - 2x^2
Profit function = -2x^2 + 198x
Now, take the derivative of the profit function and set it equal to zero:
d(profit function)/dx = -4x + 198 = 0
-4x = -198
x = 49.5
The quantity that will maximize profit is 49.5 units.
b) Finding the selling price at the optimal quantity:
To find the selling price at the optimal quantity, substitute the optimal quantity (49.5) into the demand function.
p = 400 - 21x
p = 400 - 21(49.5)
p = 400 - 1039.5
p = -639.5
The selling price at the optimal quantity is -$639.5 per unit. However, since negative prices don't make sense in this context, it is likely there is an error in the calculations or the question itself.
c) Finding the maximum profit:
To find the maximum profit, substitute the optimal quantity (49.5) into the profit function.
Profit function = -2x^2 + 198x
Profit = -2(49.5)^2 + 198(49.5)
Profit = -2(49.5)(49.5) + 198(49.5)
Profit = -4851
The maximum profit is -$4851.