Profit = Revenue - Cost (the American way and Chinese now too)

So we left off with the slope = $3.50 (which most math prep math students would have gotten by just reading the original problem, but this is a struggling student and we needed to review the slope formula anyways).

Now we have to find the "b" in the y = mx + b formula.

So we plug in the slope and we get y = 3.50x + b.

and now we plug in one of our points (0,90) or (1,93.50) to find b.

90 = 3.50(0) + b or 93.50 = 3.50(1) + b

either one should give you b = 90 !! (or what the book calls "fixed cost")

so finally we have y = 3.50x + 90 as our "Cost linear function" or C(x).

Now we need to find the Revenue function and we see that Rev = $9x.

y = 9x because we are selling the shirts at $9 each. Or if you must find the 2 points (1,9) and (2,18)...if you sell 2 shirts you collect $18 !!

slope = m = (18-9)/(2-1) = 9 and "b" of the y = mx + b is 0 because if you sell 0 shirts you collect 0 money. Or just plug in one of our points.

My student plugs in one of the points and get b = 0. Excellent !!

Now the book asks us to plot the two graphs on the same set of axes.

Are you ready to play "battleship" ? Many parents ask me how can their kids learn math. I always suggest Monopoly (adding the dice numbers real fast, counting money and learning a little bit about the real world...especially when trading properties).
Battleship is another excellent game. It teaches about the xy plane and about strategy and good old trial and error. Any card game is good too. I am sure this all translate into the computer world. Although the saddest day of my life for me was when they computerized the score keeping at the bowling alley. What are they going to do next take George Washington off the one dollar bill ? Or Lincoln off the penny ?

So basically we plot (0,90) and (1,93.50) for Cost on the paper and connect the dots.

We plot (1,9) and (2,18) for Revenue on the paper and connect those dots.

The break-even point is where these two graphs cross. Where Cost = Revenue !!

I think we have reach the climax of our tutoring session when the student stops and does not go any further. Oh my gush i think the brain is full. No, it is his cell phone vibrating and it is his mother calling for him. The hour is up and she is waiting for him in the parking lot. (we have spent an hour on one problem)

He excuses himself and runs out of the room.

He returns and says his mother is giving him one half hour more. He is excited and wants to find the Break-Even point and wants to know if Juan is going to be successful selling t-shirts because he has a friend who sells t-shirts with sayings on them. (My Honors Calculus class just had t-shirt made up that said "Nuke it" on it. Referring to the use of the graphing calculator for the midterm exam.)

So now he wants to know how to finish the problem.

So i say well we have 2 equations and 2 unknowns (we used to call these variables but that is too confusing for students, so they changed it to unknowns)

My student happily says he is going to use ADDITION to solve this problem. I suggest substitution. Oh yea, how do you do that ? As he takes the last gulp of his Pepsi and finishes his candy bar (the breakfast of champions for 90% of college students).

So i have to lead him through the process of Cost=Revenue...so 3.50x + 90 = 9x

I explain how it is like substitution in a ball game. The number 3 batter goes in for the number 3 starter. Or the point guard goes in for the starting point guard.

I am pleasantly surprise when the student does his arithmetic (with a calculator) and gets x = 16.3636363636.

Then i am totally blown away when he says Juan has to sell 17 shirt to break-even.

Mission accomplished !! and he is a good businessman. My students in high school would say...."Juan has to sell 16.363636363636 t-shirts to break even !!"

The life of the math teacher is never easy....LOL

But the problem in the book is not finish. Part "f" says we now have to find the Profit linear function.

And again my college student surprises me when he pronounces that Profit = Revenue - Cost !!
(I chime in the "American way"...and he adds "and the Chinese way too". I get a tear in my eye. This boy or man in his 20's knows so much, but so little math).

So i ask him how are we going to find this equation of Profit. And he draws a blank.

We are now back in Math class and no where to go.

I ask again what do we do with 2 equations and 2 unknowns and he says ADDITION !!
I suggest substitution.

So now we get P = 9x - 3.50x + 90

the classic biggest mistake in algebra...the negative sign

we do not have time to play around and see some wrong answers, so i just suggest that he use parenthesis in his work.

P = 9x - (3.50x + 90)

The book now asks for how many t-shirts must Juan sell to make a $500 profit.

again a complete shut down by my struggling math student.

I suggest we get a couple of points...and he figures out that selling 0 shirts will give us a profit of -90. and he proclaims, hey we lost 90 dollars. (I cherish moments like that...of course a high school student would say, you can not have negative money...i did something wrong).

His second point he says if i sell 1 shirt, we still lose money. (notice the use of the word "we", he has taken ownership of this problem of Juan's).

So now he proclaims...if we sell 17 shirts we make $500. Taking his best guess as the extra 1/2 hour is running out.

Remember 75% of the population can NOT do this problem. And if we were watching Jay Leno "street walk" or "street talk" or "street smarts"...you would say 99% of the population can NOT solve this problem including many CEO's of companies.

With time running out, i suggest putting the $500 in for the Profit.

and my student successfully finishes the problem

500 = 9x - (3.50x + 90)

x = 107.2727

and he proclaims as he gets all his stuff together and throws away his pepsi can and candy wrappers (I told you he was a good boy), I better have 108 friends if i want to make $500 selling t-shirts.

mission accomplished.

But the book now wants him to graph the profit function P(x). Oh well maybe next week.

I talk briefly with his Mom in the parking lot and she is so proud of her boy. She has to drive him to work now and he can not be late. And how do you think he will do in this math class at the college ?

I lie as usual. Some how the college and this student have gotten him this far. He will probably earn a C. He gets a bunch of points for at least trying to do his homework. They give him points for attendance and even one take home test that he should get 115 pts on. And if the students help the teacher with some surveys, they get 5 more points. A few teacher presents should seal the deal. And a lot of hard work staring at his homework and trying to take notes in class. Plus he says there is this nice Asian-Indian (thank God for the Asians) grad student who is helping him in the math lab every night. My tutoree is only taking one class per semester while he works during the day.