Stuck with differential calculus question

If the total cost (in £s) function is given by

TC = 2Q^2 + 158Q - 12000

where Q is the quantity produced

(a) What Q would minimise total costs?
Give your answer to 2 decimal places.

(b) Use your value in (a) to find the minimum value for total costs.
Give your answer to the nearest pound.

I would be extremely grateful if anyone could help me answer this question.

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Reply 1

TenOfThem

Original post by KingP1N

I would be extremely grateful if anyone could help me answer this question.

Do you know how to differentiate

If so, do so and then

\frac{dTC}{dQ} = 0

Reply 2

The Tesco Jinn

Well, you have a function, (a quadratic, a positive one at that), when is the value of the quadratic at a minimum?

How would you find this minimum?

Reply 3

KingP1N

Original post by TenOfThem

Do you know how to differentiate

If so, do so and then

\frac{dTC}{dQ} = 0

I have never been taught how to differentiate, and I am trying to teach myself. Would you mind going through how to get the answer with me?

Reply 4

TenOfThem

Original post by KingP1N

I have never been taught how to differentiate, and I am trying to teach myself. Would you mind going through how to get the answer with me?

sure

If

TC = aQ^n + bQ^m + c

then

\frac{dTC}{dQ} = anQ^{n-1} + bmQ^{m-1}

Reply 5

KingP1N

Original post by TenOfThem

sure

If

TC = aQ^n + bQ^m + c

then

\frac{dTC}{dQ} = anQ^{n-1} + bmQ^{m-1}

For question a) I got -12,000, this isn't right is it?

Reply 6

TenOfThem

Original post by KingP1N

For question a) I got -12,000, this isn't right is it?

No

Can you show what you got when you differentiated

Reply 7

KingP1N

Original post by TenOfThem

No

Can you show what you got when you differentiated

I'm not sure I'm doing it properly at all, would you be able to just go through the answers with me. Sorry if I sound thick

Reply 8

TenOfThem

Original post by KingP1N

I'm not sure I'm doing it properly at all, would you be able to just go through the answers with me. Sorry if I sound thick

I will do a different example

t = 5q^2 - 20q + 100

differentiating gives

\frac{dt}{dq} = 10q - 20

You have a minimum (or maximum for some cubes) when the differential = 0

So

10q-20 = 0

If this had been your question then q=2 would be the answer for (a) and then putting that into the original formula would give you t=80

Can you use this?

Reply 9

KingP1N

Original post by TenOfThem

I will do a different example

t = 5q^2 - 20q + 100

differentiating gives

\frac{dt}{dq} = 10q - 20

You have a minimum (or maximum for some cubes) when the differential = 0

So

10q-20 = 0

If this had been your question then q=2 would be the answer for (a) and then putting that into the original formula would give you t=80

Can you use this?

TC = 2Q^2 + 158Q - 12000

\frac{dt}{dq} = 4Q + 158 = 0

so q=0

is this correct?

(edited 11 years ago)

Reply 10

TenOfThem

Original post by KingP1N

TC = 2Q^2 + 158Q - 12000

\frac{dt}{dq} = 4Q + 158 = 0

Yes

so q=0

is this correct?

no

how do you get q = 0 from that?

Reply 11

KingP1N

Original post by TenOfThem

Yes

no

how do you get q = 0 from that?

Will you be able to show me how to the rest. I'm still confused and I would be very grateful

Reply 12

TenOfThem

Original post by KingP1N

Will you be able to show me how to the rest. I'm still confused and I would be very grateful

what does Q need to be if 4Q+158 = 0

Reply 13

KingP1N

Original post by TenOfThem

what does Q need to be if 4Q+158 = 0

-39.5 but how can the amount produced be a negative number?

Reply 14

TenOfThem

Original post by KingP1N

-39.5 but how can the amount produced be a negative number?

True

Assuming that the question you asked was accurate that does present an anomaly

But then it seems ridiculous that if we put Q=0 in we get TC as a negative

That would suggest that a profit is made if you produce no goods

Can you double check the question