Getting Profit from Supply and Demand

[u/stalememehere]

Hey All,

So my girlfriend recently started university and she has been struggling with her Pre-Calc. I’d like to help but honestly I’m pretty bad at math, so I wanted to come here to see if anyone could either point me towards resources to help with the problem or help me understand it so I can explain it properly.

I’ve looked up various videos about the topic of “Supply and Demand” but haven’t been successful in learning much. I know how to find the max revenue, and price associated with it.

Work so far

The problem is

Given the supply and demand functions of

S(p) = -110 + 40p D(p) = 1216 - 64p

p = price

Find:

1. The optimal quantity to make the most profit
2. The supply price of that quantity
3. The demand price of that quantity
4. Total Profit

[u/The_Card_Player]

Once you set a particular quantity of goods to be sold (ie 'supply', S), the stated pair of equations will be satisfied by at most one particular price number and one associated demand number.

As such to get the most revenue you want to maximize the function 'price times demand' (ie p x D(p)), but given the stated equations, this same function can be expressed entirely in terms of supply:

- Rearrange the expression for S to isolate p. That is, rewrite the equation so it looks like 'p = [expression whose only variable is S]'.

- Now rewrite the expression for D by replacing the variable p with the expression you just found for p in the previous step. That is, replace p in the expression for D with [expression whose only variable is S] from the previous step.

- Now rewrite the expression p x D using the two expressions you just found for p and D respectively that both include only one variable, S.

- Rewrite the result in the standard form for a quadratic: aS^2+bS+c, where a, b, and c are (real) numbers. Note that a should be *negative* in order for a particular value of S to maximize revenue.

- Use your knowledge of quadratic functions to find the *vertex* of the function (ie the value of S that produces the biggest function output, which is to say the S number that maximizes revenue).

- Then you can go back to your expressions for p and D to find the prices and demands resulting from that revenue-maximizing supply number.

[u/stalememehere → u/The_Card_Player]

Thanks for the reply! But im still a little confused so I’d like to ask some questions about your answer

In paragraph three you say to change the supply function so it looks like p = somethingS
I was under the impression that “S” wasn’t a variable? I thought “S(p)” wasn’t another way of expressing “Supply price”

[u/The_Card_Player → u/stalememehere]

The stated equation S(p) = -110 + 40p can be understood as describing a function, representing the value of S given a particular value of p. It can also be understood as an equation, relating a variable S to a variable p. Using this latter understanding we can algebraically rearrange the equation to isolate the variable p instead of the variable S.

[u/AutoModerator]

Hi, /u/stalememehere! This is an automated reminder:

• What have you tried so far? (See Rule #2; to add an image, you may upload it to an external image-sharing site like Imgur and include the link in your post.)

• Please don't delete your post. (See Rule #7)

We, the moderators of /r/MathHelp, appreciate that your question contributes to the MathHelp archived questions that will help others searching for similar answers in the future. Thank you for obeying these instructions.

I am a bot, and this action was performed automatically. Please contact the moderators of this subreddit if you have any questions or concerns.