Need real help! Linear Business Modeling!

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JDRhoads

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I am having trouble coming up with separate linear models? Any help would be appreciated!

The Problem looks like this:

Two people decide to open a business reconditioning toner cartridges for copy machines. They rent a building for $7000 per year and estimate that building maintenance, taxes, and insurance will cost $6500 per year. Each person wants to make $12 per hour in the first year and will work 10 hours per day for 260 days of the year. Assume that it costs $28 to restore a cartridge and that the restored cartridge can be sold for $45.

1. Write a linear function for the total cost C to operate the business and restore n cartridges during the first year, not including the hourly wage the owners wish to earn.

2. Write a linear function for the total revenue R the business will earn during the first year by selling n cartridges.

Thanks!!
 
I'm glad to read that you're willing to accept any kind of help. We're big on guiding people out of confusion, here.

Please explain what you've been able to do thus far. That is, what have you already learned about these types of exercises and at what point have you become stuck?

Please be complete, in describing your situation, so that we may determine where to begin helping you and at what level. Cheers.
 
I have looked at the problem and thought that the number 1 may be:

c(x) = 28(x) + 13,500

C(x) otherwise being Y from your perspective.

Although I am usually pretty bad at assembling linear equations and there always seems to be a variable in which is questionably different depending on the teacher you have...

When I look at a problem like this, I can read the words all I want. But it is as garbled in my mind as staring at a Korean sentence and trying to decipher what it says. Its like staring at a T.V. with a blindfold on in your mind and I can't unveil the cloud in my mind.

I have generally never been good at mathematics and will enjoy the day in life where I don't stare at these ridiculously unrealistic problems. But suffice I usually have trouble understanding now to form an equation out of linear story problems.
 
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JDRhoads said:
c(x) = 28(x) - 13,500
Click to expand...

That's a much better start than we usually get from people around here!

One comment about notation, and one error to fix:

Use the symbols that they provided: y = C(n)

C is the total cost (less wages). So we ADD the two costs (variable cost to repair n cartridges PLUS fixed costs of $13,500).

C(n) = 28n + 13500

What do you come up with for R(n) ?
 
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So the costs would be C(n) = 28n + 13,500

But to add it in terms of revenue I am assuming number 2 would be:

R(n) = 17n - 13,500

Not counting wages. But including wages I think the equation should look like :

R(n) = 17n - 75,900


I took 12 dollars an hour and multiplied it by 10 hours per day. And multiplied that by 260 days a year. And multiplied that result by 2 in order to account for the second business person. Then added that number on to the 13,500.
 
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JDRhoads said:
costs would be C(n) = 28n + 13,500

But to add it in terms of revenue I am assuming [ R(n) ] would be:

R(n) = 17n - 13,500

Not counting wages. But including wages I think the equation should look like :

R(n) = 17n - 75,900
Click to expand...

To add the costs in terms of revenue? I'm not understanding your pronoun it (above), so I don't understand that statement.

Revnue is the money that a business brings in. Wages are costs that a business pays out.

Do not add wages to revenue because wages are not a source of money for the business; they are an expense.

In the first year, the business sells n repaired cartridges for $45 dollars each.

In other words, multiply the number of cartridges by the dollar amount that they get for each.

That's all. ;)
 
There are also 4 more questions to this linear equation.

On number 3 it asks:

"How many cartridges must the business restore and sell annually to break even, not including the hourly wage the owners wish to earn? Round up to the nearest whole."

I proceeded by :

0 = 17n - 13,500
13,500 = 17n
794 = n


On number 4 it asks:

"How many cartridges must the business restore and sell annually for the owners to pay all expenses and earn the hourly wage they desire? Round to the nearest whole."

I am not sure I understand this question assuming my previous math is correct. Theoretically are they asking how much they'd have to make including the wage variables to break even? That doesn't sound very realistic for a business.
 
Just read your post. Then I don't understand how to write question 2. Because without deducting total fixed costs there is no linear equation to be made other than 17n. Which in the money received times the number of cartridges.
 
Because it costs them 28 to restore. And 45 - 28 = 17. So isn't the actual revenue 17? I don't understand why this can't simply be one equation rather than making it complicated on purpose by splitting the equation in to parts.
 
JDRhoads said:
Because it costs them 28 to restore. And 45 - 28 = 17. So isn't the actual revenue 17? I don't understand why this can't simply be one equation rather than making it complicated on purpose by splitting the equation in to parts.
Click to expand...

Seems like you're kinda "jumpin' the gun".

You're thinking "net" or "profit", but part (2) is talking about gross revenue (that is, all of the money that comes INTO the business. Don't subtract anything, yet.)

Cost is modeled in terms of the number of cartridges:

C(n) = 28n + 13500

Revenue is modeled in terms of the number of cartridges, too:

R(n) = 45n

The reason we do it this way, is to demonstrate lessons on how to begin building "bigger" functions using smaller functions as "parts".

Now, we could say that Profit is also modeled in terms of the number of cartridges because Profit is Revenue minus Costs.

In other words, we use functions R and C to create a new function P:

P(n) = R(n) - C(n)

Don't forget, we are completely ignoring wages (per instruction), at this point. :cool:
 
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JDRhoads said:
On number 3 it asks:

"How many cartridges must the business restore and sell annually to break even, not including the hourly wage the owners wish to earn? Round up to the nearest whole."
Click to expand...

Break even means no money lost but no money made, either, at the end of the year. In other words, zero profit.

P(n) = R(n) - C(n)

Replace symbols R(n) and C(n) with their respective algebraic definitions (in terms of n as shown in my previous reply).

Now simplify. You will have the P function.

Solve P(n) = 0 for n, and that will be the number of cartridges that need to be repaired and sold in one year to break even.

Can you try this now?
 
JDRhoads said:
So the costs would be C(n) = 28n + 13,500

But to add it in terms of revenue I am assuming number 2 would be:

R(n) = 17n - 13,500

Not counting wages. But including wages I think the equation should look like :

R(n) = 17n - 75,900


I took 12 dollars an hour and multiplied it by 10 hours per day. And multiplied that by 260 days a year. And multiplied that result by 2 in order to account for the second business person. Then added that number on to the 13,500.
Click to expand...
You did not explain where you are in your math education. That makes it more difficult to give you quick complete answers.

Frankly, this problem is not clearly written, and that is part of your difficulty. I suspect another part of your difficulty here is the word "revenue." It usually means the money received from selling goods and services. So wages and other expenses paid out would not be included in revenue. (In VERY old fashioned texts or very sophisticated texts, you may find a phrase such as "net revenue," which has various specialized meanings depending on context.) So if they repair n machines and get $45 for each one repaired, the overall revenue R = what?

I want to give a short pep talk on word problems in algebra. They almost always want you to find an unknown number or to specifiy a relationship between variables. FIRST, write down a brief definition for each unknown or variable and assign a unique letter to represent each one. That is one set of things that you do not have to remember. SECOND, write down in mathematical form all the things that the problem says about those unknowns or variables. You are not yet trying to do any math. You are just trying to put all the pieces of the problem in front of you in a way that lets you use the math you know. If all the problem wants is relationships you are done. If the problem wants a numerical solution, you now have a problem in pure math to solve.

In the case of this problem, they have almost done the first step for you

C = cost of operating the business (excluding wages).

R = revenue from operating the business.

n = the number of units repaired.

The problem says that the cost of repairing each unit (they mean the direct cost excluding wages; I told you the problem is badly worded) is $28, and the indirect costs are $7000 and $6500. So as mmm said you simply write that down in mathematical language using the letters previously assigned.

C = 28n + 7000 + 6500 = 28n + 13500.

Now do the same thing for R.

This gives you a method for breaking a word problem down into bite sized chunks. Define first, translate second, and solve third if that is required.
Once you learn to follow that method consistently, word problems lose a lot of their mystery. (Not all; finding the mathematical core to a problem always takes a little flash of insight.)
 
Part (4) will require deeper thinking!

Let's wait until you get the costs, revenue, and profit relationship straight, before moving on to part (4) -- which will utilize the wage info for the first time. :cool:
 
So number 1 would be C(n) = 28n + 13,500 and this "Cost"

Number 2 would be R(n) = 45n and this is "Revenue"

When we reach number 3 this is when we combine the two equations?

So P(n) = 45n - 28n - 13,500 ?
 
I am going to bed and shall leave leave you in mmm's capable hands. You are quite right that revenue = 45n.

You have been mixing up profit and revenue. Revenue is the money that comes into a business by selling goods and services. Expense is what it costs the business to provide those goods and services. Profit (or loss) is what is left from revenue after expense.

So you are absolutely right. If P = profit, P = R - C = 45n - 28n - 13500 = 17n -13500.
 
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JDRhoads said:
So number 1 would be C(n) = 28n + 13,500 and this "Cost"

Number 2 would be R(n) = 45n and this is "Revenue"

When we reach number 3 this is when we combine the two equations?

So P(n) = 45n - 28n - 13,500 ?
Click to expand...

Yup. Simplify your expression for the profit function P.

Then, substitute 0 for P(n) on the left-hand side because P(n)=0 profit at break-even point.

This gives you linear equation to solve for n.
 
So for number 3 I wrote :

0 = 45n - 28n - 13,500

0 = 17n -13,500

13,500 = 17n

794.1176 = n

794 = n cartridges sold to break even
 
Looks good.

Now, the wages come into play in part (4)...

Two people are going to work 260 days in the year.

They each want $12 per hour for their trouble.

They each work 10 hours per day.

How much money is that total?

This total is the amount of profit that the business needs to earn, in order to pay those guys.

Set P(n) = the total ANNUAL wages for both guys

Solve for n
 
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I calculated that part earlier. Working for 12 dollars an hour, 10 hours a day, for 260 days of the year comes up to 31,200 per person. 62,400 combined with both owners.

So would that be?

0 = 45n - 28n - 13,500 - 62,400

0 = 17n - 75,900

75,900 = 17n

4464.705882 = n

4465 = n
 
JDRhoads said:
0 = 45n - 28n - 13,500 - 62,400

0 = 17n - 75,900

75,900 = 17n

4464.705882 = n

4465 = n
Click to expand...

Yes. Very good result. Just some minor pointers below. (You probably understood the goals at first, but not in the way that this exercise leads you to them.)


When we create function P(n) by subtracting C(n) from R(n), we only need to do that once. In other words, you don't need to write out the derivation of the profit equation, after you determine it the first time.


Hence, for part (4), you may simply show workings as:

P(n) = 17n - 13500

62400 = 17n - 13500

17n = 75900

n = 75900/17 ~= 4464.7

"They need to repair and sell 4,465 cartridges to realize their desired annual income."


Cheers :cool:
 
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