Question

Hello :)For a certain company, the cost function for producing x items is C(x)=40x+100 and the revenue function for selling x items is R(x)=−0.5(x−90)^2+4,050 . The maximum capacity of the company is 150 items.The profit function P(x) is the revenue function R(x) (how much it takes in) minus the cost function C(x) (how much it spends). In economic models, one typically assumes that a company wants to maximize its profit, or at least make a profit!1. Assuming that the company sells all that it produces, what is the profit function?2. What is the domain of P(x) ?3.The company can choose to produce either 50 or 60 items. What is their profit for each case, and which level of production should they choose? Profit when producing 50 items? Profit when producing 60 items?4. Can you explain, from our model, why the company makes less profit when producing 10 more units?

Answer 1

For the exercise, we have information about the cost function and the revenue function.

1.We find the profit function by subtracting the revenue function from the cost function. Thus,

`\begin{gathered} P(x)=R(x)-C(x) \\ P(x)=(−0.5\left(x−90\right)^2+4050)-(40x+100) \\ P(x)=-0.5\left(x-90\right)^2+4050-\left(40x+100\right) \\ P(x)=-0.5\left(x^2-180x+8100\right)+4050-\left(40x+100\right) \\ P(x)=-0.5x^2+90x-4050+4050-\left(40x+100\right) \\ P(x)=-0.5x^2+90x-4050+4050-40x-100 \\ P(x)=-0.5x^2+50x-100 \end{gathered}`

The profit function is P(x)= -0.5x^2+50x-100.

2. Since x indicates the number of items is a number that cannot be negative and since the maximum production capacity of the company is 150 the domain of the profit function is determined by the values that x can take, i.e. the domain values between 0 and 150. Using interval notation [0,150] and using set notation

`\lbrace x:0\leq x\leq150\rbrace`

3. In order to know the profit in both production cases, we evaluated the profit function at 50 and 60 as follows

`\begin{gathered} P(50)=-0.5(50)^2+50(50)-100=1150 \\ P(60)=-0.5(60)^2+50(60)-100=1100 \\ \end{gathered}`

Taking into account the above result, the company should produce 50 items to obtain a higher profit.

-The Profit when producing 50 items is 1150

-The Profit when producing 50 items is 1100

4. If we look at the profit function, this is a negative quadratic function, so it represents graphically a parabola that opens downward and in that order of ideas, the maximum profit point will be at the vertex of the parabola. The vertex is (50,1150), which is why from that point the benefits begin to be lower.