Answer:
The answer is below
Step-by-step explanation:
When a commodity is sold for x dollars per item, the number of items sold is f(x). The item costs 3 dollars to make. (For all of the below, the answer has been partially filled in. Do these problems on paper then simplify your answer until it can be used to fill in what is below. Leave no box blank)
(a)Express the total profit P in terms of x (the answer has been partially filled in for you). P = _______*f(x)
(b)What is the rate of change of the total profit as x changes? (Do this on paper then fill in the below. Leave neither box blank) ______*f(x) + _______*f '(x)
(c)If 3000 items sell when the price is 45 dollars and if the number of items that are sold decreases by 600 for every 1 dollar increase in price, find the rate of change of the profit when the item price is 45 dollars. _____________
Solution:
The cost of producing the items is the total expenses of production while the revenue is the amount of money generated from selling the item.
a) Cost = cost of producing one item * number of items produced
Cost = $3 * f(x) = 3f(x)
Revenue = selling price of one item * number of items produced
Revenue = x * f(x) = xf(x)
Profit = Revenue - Cost = xf(x) - 3f(x)
Profit = (x - 3)f(x)
b) P' = d/dx(xf(x) - 3f(x) )
P' = f(x) + xf'(x) - f'(x)
P' = f(x) + (x - 1) f'(x)
c) Given that f(x) = 3000, x = $45, f'(x) = -600 items / 1$
Substituting gives:
P' = 3000 + (45 - 1)(-600)
P' = 3000 + 26400
P' = 29400
