The total cost (in dollars) for a company to manufacture and sell
x items per week is C(x)=70x+400. If the revenue brought in by selling all
x items is R(x)=90x−0.05x^2, find the weekly profit. How much profit will be made by producing and selling 60 items each week? Hint:P(x)=R(x)−C(x)
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To find the weekly profit, we need to subtract the cost function (C(x)) from the revenue function (R(x)).
P(x) = R(x) - C(x)
Given that:
C(x) = 70x + 400
R(x) = 90x - 0.05x^2
First, substitute the values of C(x) and R(x) in the profit function:
P(x) = (90x - 0.05x^2) - (70x + 400)
Simplify:
P(x) = 90x - 0.05x^2 - 70x - 400
Combine like terms:
P(x) = -0.05x^2 + 20x - 400
Now we can calculate the profit for producing and selling 60 items each week:
P(60) = -0.05(60)^2 + 20(60) - 400
Simplify:
P(60) = -0.05(3600) + 1200 - 400
P(60) = -180 + 1200 - 400
P(60) = 840
Therefore, the weekly profit made by producing and selling 60 items each week is $840.
To find the weekly profit, we need to subtract the total cost (C(x)) from the revenue (R(x)).
The profit function, P(x), can be calculated as follows:
P(x) = R(x) - C(x)
Given that C(x) = 70x + 400 and R(x) = 90x - 0.05x^2, we can substitute these values into the profit function:
P(x) = (90x - 0.05x^2) - (70x + 400)
Simplifying the equation:
P(x) = 90x - 0.05x^2 - 70x - 400
Combining like terms:
P(x) = -0.05x^2 + 20x - 400
Now we have the expression for the weekly profit, P(x).
To find the profit made by producing and selling 60 items each week, we substitute x = 60 into the profit function:
P(60) = -0.05(60)^2 + 20(60) - 400
Calculating:
P(60) = -0.05(3600) + 1200 - 400
P(60) = -180 + 1200 - 400
P(60) = 620
Therefore, the profit made by producing and selling 60 items each week is $620.
Follow the hint and simplify
P(x)=R(x)−C(x)
then replace x with 60 and evaluate