Assuming that the company sells all that it produces, what is the profit function?

Question

For a certain company, the cost function for producing x items is C(x)=40x+200 and the revenue function for selling x items is R(x)=−0.5(x−120)2+7,200. The maximum capacity of the company is 160 items.

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 80 or 90 items. What is their profit for each case, and which level of production should they choose?
4. Profit when producing 80 items =    
5. Profit when producing 90 items =   
6. Can you explain, from our model, why the company makes less profit when producing 10 more units?

 

Answer 1

1. Profit function P(x) = R(x ) - C(x) = − 0.5(x−120)² + 7200 - 40x - 200 = - 0.5x² + 80x - 200
2. Domain is x ≥ 0
3. P(80) = $3000,
4. P(90) = $2950 So, better to have production level x = 80 units
5. Marginal profit function is P'(x) = - x + 80 = 0, x = 80 is maximum value of P(x) because P''(80) = -1 < 0.
6. So, P(90) it will be less than P(80)

Answer 2

Hi! I hope this helps answer some of your questions. I am assuming the 2 following (x-120) is in fact (x-120)².

First up, you're going to want to expand and simplify the revenue function. To do that first expand your bracket.

1. (x-120)² = (x-120)(x-120)
2. Expand using FOIL. You will get (x² - 120x - 120x + 14400)
3. Simplify to get (x² - 240x + 14400)
4. Lets put that expanded form back in our equation:
5. -0.5 (x² - 240x + 14400) + 7200
6. Now multiply out the bracket by -0.5
7. -0.5x² + 120x -7200 + 7200
8. Note that the -7200 +7200 cancel to make 0
9. We end up with our revenue function being: -0.5x² + 120x
10. To find the profit function we need to do the revenue function subtract the cost function.
11. -0.5x² + 120x - (40x + 200)
12. We expand to get: -0.5x² + 120x - 40x - 200
13. Now simplify: -0.5x² + 80x - 200
14. This is our profit function.

For the Domain

The question tells us the most we items (x) we can make is 160. We can't make negative items so the least is 0.

This can be written as 0 ≤ x ≤ 160 or in interval notation [0,160]

For the final components of the question we need to sub 80 and 90 into our x's in our profit function.

1. -0.5(80)² + 80(80) - 200
2. 3000
3. -0.5(90)² + 80(90) - 200
4. 2950
5. From those two levels of production the greatest profit is achieved when 80 items are produced, so this is the level of production they should opt for.

For the last stage, graphing the equation can help but you can also do a quick check by subbing in x=79 and x=81 to see that x is the maximum point of the graph, meaning it yields the highest possible profit. Also note that the leading coefficient is negative so the x² curve increases first then decreases (like an upside down u).

I hope that answers your questions!