I'm working on a project and I need some help with figuring out the equations to write.
Here is all the information about the project:
You and your friends decide to produce your own CD. After searching on the Internet, you find a low-cost CD duplication outfit that will let you upload your audio files and graphics over the internet, and then burn them onto CDs, print program booklets, assemble, and shrink-wrap the product. The company, Mixup LLC, sends you their pricing information. In the meantime, your friends go to concerts and find out about typical CD sales at the types of venues you play. Your goal is to figure out how many CDs you should produce to make as much profit as possible.
The facts are:
·You have a friend with computer and audio equipment who is willing to produce all six of your tracks for $50 cash.
·You will make the artwork yourself, for free.
·Mixup charges $13.80 to set up the CD burning process and $5.35 per CD made.
·A sampling of recent concert shows that at a price of $15 each, only 20 CDs were sold. At a price of $10 each, 60 CDs were sold.
Use this information to come up with a mathematical model-in this case, a function giving the overall profit in terms of the number of CDs, and then find the number of CDs that produces the absolute maximum profit.
I was thinking that it may go something like (50.00 +13.80)-5.35x. Am I on the right track?
Yes, you are on the right track. To come up with a mathematical model for the overall profit, we need to consider all the costs involved and the revenue generated from selling the CDs.
Let's break down the costs and revenue:
Costs:
1. Friend's fee for producing tracks: $50
2. Mixup's setup fee: $13.80
3. Mixup's cost per CD: $5.35
Revenue:
1. Revenue per CD sold: We have information about CD sales at different prices. At $15 price, 20 CDs were sold, and at $10 price, 60 CDs were sold.
Based on this information, we can make the following assumptions:
1. The demand for CDs is linearly related to the price per CD (assuming other factors are constant).
2. The revenue is equal to the number of CDs sold multiplied by the price per CD.
Mathematically, we can represent the revenue as:
Revenue = Number of CDs sold * Price per CD
The total cost can be calculated as:
Total Cost = Friend's fee for producing tracks + Mixup's setup fee + Mixup's cost per CD * Number of CDs
The profit is then given by:
Profit = Revenue - Total Cost
To find the number of CDs that produces the absolute maximum profit, we need to determine the value of x (number of CDs) that maximizes the profit function.
Putting it all together, the mathematical model for the overall profit in terms of the number of CDs (x) can be written as:
Profit(x) = x * Price per CD - (Friend's fee + Mixup's setup fee + Mixup's cost per CD * x)
To find the number of CDs that produces the absolute maximum profit, you can differentiate the profit function with respect to x, set the derivative equal to zero, and solve for x. The resulting value of x will give you the optimal number of CDs to produce.
Keep in mind that the assumptions made in this model may not perfectly reflect real-world scenarios, but they provide a starting point for analysis and decision-making.