A manufacturer estimates that when x units of a particular commodity are
produced each month, the total cost (in dollars) will be
C(x) = 1/8x^2+4x+200
and all units can be sold at a price of p(x) = 49 - x dollars per unit. Determine the price that corresponds to the maximum profit
Yes
To determine the price that corresponds to the maximum profit, we need to find the profit function and then find the maximum of that function.
The profit for selling x units is given by the equation:
Profit(x) = Revenue(x) - Cost(x)
To calculate the revenue, we multiply the number of units sold (x) by the selling price per unit (p(x)):
Revenue(x) = x * p(x)
Substituting the given equation for p(x):
Revenue(x) = x * (49 - x)
Now let's find the equation for the cost function using the given equation:
C(x) = 1/8x^2 + 4x + 200
To obtain the profit function, we subtract the cost from the revenue:
Profit(x) = Revenue(x) - Cost(x)
Profit(x) = x * (49 - x) - (1/8x^2 + 4x + 200)
Simplifying the expression for profit:
Profit(x) = x * (49 - x) - 1/8x^2 - 4x - 200
Now we have the profit function in terms of x. To find the price that corresponds to the maximum profit, we need to find the value of x that maximizes the profit function.
To find the maximum value, take the derivative of the profit function with respect to x and set it equal to zero. Then solve for x to find the critical point(s). Once you have the critical point(s), evaluate the profit function at those point(s) to find the maximum profit.
Let's calculate the derivative of the profit function:
d/dx [Profit(x)] = d/dx [x * (49 - x) - 1/8x^2 - 4x - 200]
Using the power rule and simplifying:
d/dx [Profit(x)] = 49 - 2x - 1/4x - 4
Setting the derivative equal to zero and solving for x:
49 - 2x - 1/4x - 4 = 0
Combining like terms:
-2x - 1/4x = -45
Multiplying both sides by -4 to eliminate the fraction:
8x + x = 180
Combining like terms:
9x = 180
Dividing both sides by 9:
x = 20
Now we have the critical point x = 20.
To find the maximum profit, substitute x = 20 into the profit function and evaluate:
Profit(20) = 20 * (49 - 20) - 1/8(20^2) - 4(20) - 200
Profit(20) = 20 * 29 - 400 - 80 - 200
Profit(20) = 580 - 400 - 80 - 200
Profit(20) = -100
The maximum profit occurs at x = 20, and the maximum profit is -100 dollars.