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When a commodity is sold for x dollars per item, the number of items sold is f(x). The item costs 5 dollars to make. The item costs 3 dollars to make.

Required:
a. Express the total profit P in terms of x (the answer has been partially filled in for you).
b. What is the rate of change of the total profit as x changes?
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.

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Lhmarianateixeiraid

In this exercise we will use the differential equations to calculate finances, we find that:

a) [tex]Profit = (x - 3)f(x)[/tex]

b) [tex]P' = f(x) + (x - 1) f'(x)[/tex]

c) [tex]P' = 29400[/tex]

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)Then using the given formula and substituting for the known values ​​we find that;

[tex]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)[/tex]

b) Applying knowledge of differential equations we find that:

[tex]P' = d/dx(xf(x) - 3f(x) )\\P' = f(x) + xf'(x) - f'(x)\\P' = f(x) + (x - 1) f'(x)[/tex]

c)Substituting the known values ​​into the formula given above, we find that:

[tex]f(x) = 3000, x = $45, f'(x) = -600 items / 1$\\P' = 3000 + (45 - 1)(-600)\\P' = 3000 + 26400\\P' = 29400[/tex]

See more about differential equations at brainly.com/question/25731911

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