Trouble with some profit, revenue, cost questions

[conp]

I am having trouble with a few questions that deal with cost, profit, revenue, avg. cost, marginal cost. I have attempted a few but others I am a little lost.

1)
Total cost of producing q units of a product is given by c(q)=q^3-60q^2+1400q+1000 for 0<(eq.)q<(eq)50; The product sells for $788 per unit. What production level maximizes profit? Find the total cost, total revenue, and total profit at this production level. Where is profit maximized.

I am completely lost. I know MC=MR and profit=R(q)-C(q), but thats it.

2)The total cost C(q) of producing q goods is given by:
C(q)=0.01q^3-0.6q^2+13q.
a)What is fixed cost? I assumed since each number is being multiplied by a variable, there is no fixed cost. Actual answer is 0.
b) What is the maximum profit if each item is sold for $7?
I did c(q)=0.01(q^3)-0.6^2+13q
plug in 7 for q
3.43+.36+91=94.79. Actual answer is 96.56.
c)If 34 goods are produced. They all sell when price is 7 each, but for each 1 increase in price, 2 fewer goods are sold. What should the price be sold at?
Actual answer is $5.

3) Let c(q)=0.04q^3-3q^2+75q+96 is the tot. cost of producing q items.
a)Find the avg. cost per item as a function of q.
b)what values of q is the avg. cost per item decreasing? increasing?
c)what value of q is the avg. cost per item smallest?

My answer: a)a(q)=c(q)/q=0.04q^2-3q+75+96
a'(q)=0.08q-3
0=0.08q-3
d)q=37.5

 

[JeffM]

Re: Trouble with somer profit, revenue, cost questions

I am having trouble with a few questions that deal with cost, profit, revenue, avg. cost, marginal cost. I have attempted a few but others I am a little lost.

1)
Total cost of producing q units of a product is given by c(q)=q^3-60q^2+1400q+1000 for 0<(eq.)q<(eq)50; The product sells for $788 per unit. What production level maximizes profit? Find the total cost, total revenue, and total profit at this production level. Where is profit maximized.

I am completely lost. I know MC=MR and profit=R(q)-C(q), but thats it.

So, if the price is $788 per unit, what is the equation for total revenue?
Do you know how to find marginal revenue from the equation for total revenue?
Do you know how to find the marginal cost from the equation for total cost?
In general it is not true that marginal cost = marginal revenue at every quantity of sales. The quantity at which marginal cost equals marginal revenue is the profit maximizing quantity. So you need to find where the expressions for marginal cost and marginal revenue are equal. Once you have that you substitute back into the equations for R and C.
See how far these hints take you.

2)The total cost C(q) of producing q goods is given by:
C(q)=0.01q^3-0.6q^2+13q.
a)What is fixed cost? I assumed since each number is being multiplied by a variable, there is no fixed cost. Actual answer is 0. So you are right and for the right reason.

b) What is the maximum profit if each item is sold for $7?
I did c(q)=0.01(q^3)-0.6^2+13q You said above that you are to equate MARGINAL revenue and MARGINAL cost. This is an equation for TOTAL cost, right?

plug in 7 for q Is $7 a quantity or a price? If it is a price why are you substituting a price for quantity in the cost equation?

3.43+.36+91=94.79. Actual answer is 96.56. You have to find the quantity that maximizes profit by equating marginal cost and marginal revenue. Then calculate total revenue and total cost at that quantity, and then calculate the profit.

c)If 34 goods are produced. They all sell when price is 7 each, but for each 1 increase in price, 2 fewer goods are sold. What should the price be sold at?
What should the price be sold at? That cannot be the actual problem statement. I cannot give even a hint because I do not understand the question. Actual answer is $5.

3) Let c(q)=0.04q^3-3q^2+75q+96 is the tot. cost of producing q items.
a)Find the avg. cost per item as a function of q.
b)what values of q is the avg. cost per item decreasing? increasing?
c)what value of q is the avg. cost per item smallest?

My answer: a)a(q)=c(q)/q=0.04q^2-3q+75+96 Hmm You divided everything by q except for the constant term of 96. Does this make sense now that you think about it?

a'(q)=0.08q-3
0=0.08q-3
d)q=37.5

 

[conp]

Re: Trouble with somer profit, revenue, cost questions

Thanks for the response.

I'm pretty sure total revenue is R(q)=P*Q

1)Would you set q^3-60q^2+1400q+1000=788
and then solve for q. Once you get q, you would plug it back into the equation and get total cost?

2) b) That is Total cost. Marginal Cost is C'(q) so its just the derivative of C(q). 7 is the price of each item sold. We are looking at the maximum profit if each item were sold at that amount. You say to equate MR and MC. So are you saying MR=MC and solve for p, or have profit=R(q)-C(q)?

c) is the original equation just assume that 34 goods are produced. q=34. p=7, but each $1 increase causes 2 fewer goods to be sold. What should the the price be raised to?

I am having trouble with a few questions that deal with cost, profit, revenue, avg. cost, marginal cost. I have attempted a few but others I am a little lost.

1)
Total cost of producing q units of a product is given by c(q)=q^3-60q^2+1400q+1000 for 0<(eq.)q<(eq)50; The product sells for $788 per unit. What production level maximizes profit? Find the total cost, total revenue, and total profit at this production level. Where is profit maximized.

I am completely lost. I know MC=MR and profit=R(q)-C(q), but thats it.

So, if the price is $788 per unit, what is the equation for total revenue?
Do you know how to find marginal revenue from the equation for total revenue?
Do you know how to find the marginal cost from the equation for total cost?
In general it is not true that marginal cost = marginal revenue at every quantity of sales. The quantity at which marginal cost equals marginal revenue is the profit maximizing quantity. So you need to find where the expressions for marginal cost and marginal revenue are equal. Once you have that you substitute back into the equations for R and C.
See how far these hints take you.

2)The total cost C(q) of producing q goods is given by:
C(q)=0.01q^3-0.6q^2+13q.
a)What is fixed cost? I assumed since each number is being multiplied by a variable, there is no fixed cost. Actual answer is 0. So you are right and for the right reason.

b) What is the maximum profit if each item is sold for $7?
I did c(q)=0.01(q^3)-0.6^2+13q You said above that you are to equate MARGINAL revenue and MARGINAL cost. This is an equation for TOTAL cost, right?

plug in 7 for q Is $7 a quantity or a price? If it is a price why are you substituting a price for quantity in the cost equation?

3.43+.36+91=94.79. Actual answer is 96.56. You have to find the quantity that maximizes profit by equating marginal cost and marginal revenue. Then calculate total revenue and total cost at that quantity, and then calculate the profit.

c)If 34 goods are produced. They all sell when price is 7 each, but for each 1 increase in price, 2 fewer goods are sold. What should the price be sold at?
What should the price be sold at? That cannot be the actual problem statement. I cannot give even a hint because I do not understand the question. Actual answer is $5.

3) Let c(q)=0.04q^3-3q^2+75q+96 is the tot. cost of producing q items.
a)Find the avg. cost per item as a function of q.
b)what values of q is the avg. cost per item decreasing? increasing?
c)what value of q is the avg. cost per item smallest?

My answer: a)a(q)=c(q)/q=0.04q^2-3q+75+96 Hmm You divided everything by q except for the constant term of 96. Does this make sense now that you think about it?

a'(q)=0.08q-3
0=0.08q-3
d)q=37.5

 

[JeffM]

Re: Trouble with somer profit, revenue, cost questions

Thanks for the response. You're welcome. Let's try one question at a time OK? I'm pretty sure total revenue is R(q)=P*Q. Absolutely correct, but please keep your notation consistent: R(q) = Pq. Price is units of money per widget sold so to find the the total money received you just multiply the unit price times the number of units sold. So, IN THIS CASE, MR = dR/dq = P = 788.

1)Would you set q^3-60q^2+1400q+1000=788 NO. You are equating marginal revenue and TOTAL cost.
and then solve for q. You do solve for q but by equating marginal cost and marginal revenue. Once you get q, you would plug it back into the equation and get total cost? Yes, plus you can get marginal cost, average cost, total revenue, and profit because they are all expressed in terms of q and P.

So now let's do this question. Then see how far you can get with the next.

 

[conp]

So if MR=dR/dq=p=788 then we have p. Now all we need is to find q. To find q would you take the derivative of C(q). C'(q)=3q^2-120q+1400? Then set is equal to zero and solve?

 

[JeffM]

So if MR=dR/dq=p=788 then we have p.

You had price all along. What you did not have was marginal revenue. In this case, which is the standard case for a firm in a perfectly competitive industry, marginal revenue equals price, but it does not for other types of firm. So, the profit maximizing formula GENERALLY requires you to find marginal revenue in terms of p or q. In the SPECIAL case of a firm in a perfectly competitive industry, marginal revenue equals unit price.

Now all we need is to find q. Correct

To find q would you take the derivative of C(q). C'(q)=3q^2-120q+1400? Then set is equal to zero and solve?

No. Let's start from the beginning. Let Y(q) = profit as a function of q.
Y(q) = R(q) - C(q), right? Revenue minus cost is profit. Simple identity.
To maximize Y(q), take ITS derivative and set equal to zero.
So q such that Y`(q) = 0 gives you the profit maximizing quantity. Basic calculus.
But Y(q) = R(q) - C(q).
So Y`(q) = R`(q) - C`(q). Again, basic calculus.
So, when Y'(q) = 0, R`(q) = C`(q). Basic algebra.
You do NOT set C`(q) = 0. You set it equal to R`(q). And in this case, you found that R`(q) = 788.
Now solve the problem. If you understand this one, you will get the others readily.