Final answer:
To find the maximum profit for a given demand price and total cost function, calculate the profit function, determine its derivative, and solve for the quantity that results in a zero derivative. Substituting this quantity into the profit function will give the maximum profit value, which calculates to a maximum profit of approximately $267.30.
Step-by-step explanation:
To determine the maximum value of the profit for a product with a demand price of $18, and total costs given by the function c(x)=0.8x² - 3.6x + 9, we must first calculate the total revenue and then find the difference between the total revenue and the total costs.
Total revenue (TR) is calculated by multiplying the demand price by the quantity sold (x), so TR = $18x.
The profit function (P(x)) is then given by the total revenue minus the total costs function,
P(x) = TR - c(x) = $18x - (0.8x² - 3.6x + 9).
To find the quantity (x) that will maximize the profit, we can set the first derivative of the profit function equal to zero and solve for x (since the profit function is a parabola opening downwards, the turning point will be a maximum). Upon finding the value of x that gives the maximum profit, we can substitute this x back into the profit function to find the maximum profit value. The maximum profit value should be rounded to the nearest cent as requested.
Applying these steps, the profit function simplifies to P(x) = -0.8x² + 21.6x - 9.
Taking the first derivative, P'(x) = -1.6x + 21.6.
Setting P'(x) to zero, we get -1.6x + 21.6 = 0, solving which gives x = 13.5.
Thus, the maximum profit occurs when 13.5 units are produced.
Substituting x = 13.5 into the profit function,
P(13.5) = -$15.3 + $291.6 - $9,
which calculates to a maximum profit of approximately $267.30.