Cost Function and Profit Function: C(x) = 15x + 20,000 for x units produced

[HB09]

I am SO STUCK! I do not have a book or any examples as to how to solve this and I am so confused. I have tried Youtube videos but nothing is similar. Does anyone have an example of how to solve this? Mabe a formula to follow? Not looking for answers Im looking for help!!

Cost Function: C(x) = 15x + 20,000Where x is measured in units produced, and C(x) is measured in dollars.Find the total cost if the firm producesa) 0 units (Does this answer make sense?)
(b) 1,000 units
(c) 5,000 units
(d) 10,000 units(e) What does this function tell us about the production process? (hint: What does the algebraic structure of this function and our “mental algorithm” tell us about the variable cost and the fixed cost?)
(f) How many units can be produced for a total cost of $182,000?-------------------------------------------------------

Profit Function: P(x) = 8x − 30,000Where x is measured in units produced and sold, and P(x) is measured in dollars.What is the profit if the firm produces and sells:
(a) 0 units (Does this answer make sense?)
(b) 3,000 units
(c) 4,000 units
(d) 10,000 units
(e) Find the break–even point for this production process. (hint: Solve P(x) = 0)(f) How many units must be produced and sold to achieve a profit of $70,000?

 

[JeffM]

I am SO STUCK! I do not have a book or any examples as to how to solve this and I am so confused. I have tried Youtube videos but nothing is similar. Does anyone have an example of how to solve this? Mabe a formula to follow? Not looking for answers Im looking for help!!

Cost Function: C(x) = 15x + 20,000Where x is measured in units produced, and C(x) is measured in dollars.Find the total cost if the firm producesa) 0 units (Does this answer make sense?)
(b) 1,000 units
(c) 5,000 units
(d) 10,000 units(e) What does this function tell us about the production process? (hint: What does the algebraic structure of this function and our “mental algorithm” tell us about the variable cost and the fixed cost?)
(f) How many units can be produced for a total cost of $182,000?-------------------------------------------------------

Profit Function: P(x) = 8x − 30,000Where x is measured in units produced and sold, and P(x) is measured in dollars.What is the profit if the firm produces and sells:
(a) 0 units (Does this answer make sense?)
(b) 3,000 units
(c) 4,000 units
(d) 10,000 units
(e) Find the break–even point for this production process. (hint: Solve P(x) = 0)(f) How many units must be produced and sold to achieve a profit of $70,000?

What is a function? Being very simplistic, it usually gives an unambiguous formula involving one or more variables that equal another variable. (This is not a rigorous definition.)

\(\displaystyle C(x) = c = \text {cost in dollars, where x is the number of units produced.}\)

All that the C(x) notation tells you is that you can find c if you know x.

\(\displaystyle C(x) = 15x + 20000 \implies c = 15x + 20000.\)

Now can you solve part 1?

See how far you can get with part 2. It is just another function (formula).

 

[HB09]

What is a function? Being very simplistic, it usually gives an unambiguous formula involving one or more variables that equal another variable. (This is not a rigorous definition.)

\(\displaystyle C(x) = c = \text {cost in dollars, where x is the number of units produced.}\)

All that the C(x) notation tells you is that you can find c if you know x.

\(\displaystyle C(x) = 15x + 20000 \implies c = 15x + 20000.\)

Now can you solve part 1?

See how far you can get with part 2. It is just another function (formula).

8x-30,000? for part 2 and then fill in unit amounts?

 

[JeffM]

8x-30,000? for part 2 and then fill in unit amounts?

yes, but the problem is more than that.

In part 1, a through d just ask you to compute total cost for different numbers of units produced. Purely mechanical. Part e asks you to think about what the function means. It requires a bit of thought but no computation. Part f does not ask you to find a cost; it asks you to find the number of units produced at a given cost. How do you proceed there?

In part 2, again a through d simply though ask you to compute profits for different quantities of units produced and sold. Question a also asks whether the model makes sense. Let's put that aside for now. Questions e and f do not ask you to find the profit, but ask you to find the quantities that generate specific profits.

 

[HB09]

yes, but the problem is more than that.

In part 1, a through d just ask you to compute total cost for different numbers of units produced. Purely mechanical. Part e asks you to think about what the function means. It requires a bit of thought but no computation. Part f does not ask you to find a cost; it asks you to find the number of units produced at a given cost. How do you proceed there?

In part 2, again a through d simply though ask you to compute profits for different quantities of units produced and sold. Question a also asks whether the model makes sense. Let's put that aside for now. Questions e and f do not ask you to find the profit, but ask you to find the quantities that generate specific profits.

Thank you! I think I know how to answer all the questions now my only question is do you know what P(x) = 0) is? Im unsure what the formula is question 2 part e.

 

[JeffM]

Thank you! I think I know how to answer all the questions now my only question is do you know what P(x) = 0) is? Im unsure what the formula is question 2 part e.

I am guessing that you are taking a beginning course in micro-economics without a strong background in math. You will find that modern economics is very dependent on math. Be warned.

You still are not getting the idea behind function notation. We are giving a brief name to a formula (more exactly a rule). So instead of writing "profit in dollars" = some formula, we write P(__) = some formula. You could just as well say the variable p = the same formula. The sole advantage of writing P(x, y) is that it reminds us of what variables are used in the formula. What is the point, why not just write the formula? The point is that we may not know the formula, or it may be long and messy to write out. So this exercise is just to get you used to the idea that a function results in a number once the "arguments," meaning the variables in parentheses, are numerically specified.

So P(x) = 0 just means that the formula equals zero, and you are to solve for x. Basic first year algebra.

 

[HB09]

I am guessing that you are taking a beginning course in micro-economics without a strong background in math. You will find that modern economics is very dependent on math. Be warned.

You still are not getting the idea behind function notation. We are giving a brief name to a formula (more exactly a rule). So instead of writing "profit in dollars" = some formula, we write P(__) = some formula. You could just as well say the variable p = the same formula. The sole advantage of writing P(x, y) is that it reminds us of what variables are used in the formula. What is the point, why not just write the formula? The point is that we may not know the formula, or it may be long and messy to write out. So this exercise is just to get you used to the idea that a function results in a number once the "arguments," meaning the variables in parentheses, are numerically specified.

So P(x) = 0 just means that the formula equals zero, and you are to solve for x. Basic first year algebra.

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