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- In this video, I want to do a couple more word problems
- dealing with graphs of lines.
- So here we have a gym is offering
- a deal to new members.
- Customers can sign up by paying a registration fee of
- $200 and then a monthly fee of $39.
- This is registration.
- How much will this membership cost a member by
- the end of the year?
- So let's figure out an equation that determines how
- much total we will pay.
- p is equal to the amount that we're going to pay in total
- for our membership.
- So no matter how many months we use the gym, just to start
- using the gym, we have to pay $200 in registration.
- I'll just write-- everything we'll assume is in dollars
- --so I'll write $200.
- And then we're going to have to pay $39 for every month
- we're there.
- So then we're going to take the number of months we're
- there and multiply that times 39.
- Notice if we stay there 1 month, we'll have to pay 1
- month times $39.
- And we would have already paid the $200 registration fee.
- So it'll be $239.
- If we stay 2 months, we pay the $200 registration fee and
- then we pay 39 times 2 months, which is what?
- Like 78 or something.
- So it would be $278.
- So just to tie this altogether with linear equations and
- graphs of them, let's graph this relation.
- Remember the graph of a line can be y is
- equal to mx plus b.
- That's one of the forms. So to put this line in this form or
- this equation in this form, we can just rearrange the 39m and
- the 200 and you get p is equal to 39m plus 200.
- So what's the slope and what's the y-intercept?
- You might get confused and say, hey there's an x there
- and a y, but now you're doing it with p's and m's.
- Just remember this is the independent variable and this
- is the dependent variable.
- Here this is the independent variable.
- How many months?
- You pick a number of months and I'll tell you what the
- total cost of your membership is going to be.
- It's the same thing.
- This is like the x right there.
- This is like the y just like that.
- Just using our pattern match, this right here is the-- we
- could say it's the vertical intercept or the p-intercept
- or the-- I'm tempted to call it the y-intercept.
- But we're really intersecting the p-axis instead of the
- y-axis there.
- This right here is our slope.
- So let's graph this function.
- I won't do it too accurately.
- I just want to do a hand drawn graph just
- to give you an idea.
- We could just stay in the first quadrant.
- We're not going to stay negative months and the gym is
- never going to pay us money.
- So right off the bat, we're going to have
- to pay the gym $200.
- $200 for 0 months.
- Then for every month, we're going to have to
- spend an extra $39.
- So the slope is 39.
- Let's say this is 1 month right there.
- This is in months.
- And this axis is price, the p-axis.
- So this is like the p-intercept or the
- y-intercept.
- So after 1 month, how much are we going to have to pay?
- Well our slope is 39, so if we move 1 month forward, we're
- going to go up by 39.
- So this will, right here, that will be 239.
- If we go another month, it'll be 278.
- This is kind of a weird labeling of an axis, but I
- think you get the idea.
- So the graph of how much it'll cost us as per month will look
- something like this.
- So they say, how much will a membership cost by
- the end of the year?
- 12 months.
- We would have to go 2, 3, all the way out to 12 months,
- which might be here.
- So then our graph is going to be out here someplace.
- But we could just figure it out algebraically.
- At the end of the year, m will be equal to 12.
- When m is equal to 12, how much is our membership?
- The price of our membership is going to be our $200
- membership fee plus 39 times the number of
- months, times 12.
- What's 39 times 12?
- 2 times 9 is 18.
- 2 times 3 is 6 plus 1 is 70.
- I have a 0.
- 1 times 9 is 9.
- 1 times 3-- we want to ignore this.
- 1 times 3 is 3.
- So we have 8.
- 7 plus 9 is 16.
- 1 plus 3 is 4.
- So the price of our membership is $200 plus 39 times 12,
- which is $468.
- So it's equal to $668 at the end of our year.
- So if you went all the way out to 12, you would have to plot
- 668 someplace here on our line, if we just
- kept going out there.
- Let's do one more of these.
- Bobby and Petra are running a lemonade stand and they charge
- $0.45 for each glass of lemonade.
- In order to break even, they must make $25.
- How many glasses of lemonade must they sell to break even?
- So let me just do it with y and x.
- y is equal to the amount they make.
- Not max-- the amount they make.
- Let x is equal to the number of glasses they sell.
- What is y as a function of x?
- So y is equal to-- well for every glass they sell, they
- get $0.45 --so it's equal to $0.45 times
- the number of glasses.
- There's not any kind of minimum fee that they need to
- charge or they don't say any kind of minimum cost that they
- have to spend to run this place.
- How much in order to break even for
- each a glass of lemonade?
- They need to make $25.
- So in order to break even, they must make $25.
- So how many glasses of lemonade do they need to sell?
- y needs to be equal to $25.
- How many glasses do they need to sell?
- Well you just set this equation.
- You say 0.45x has to be equal to 25.
- We can divide both sides by 0.45.
- On the left-hand side, you're just left with an x.
- You get x is equal to-- What is 25 divided by 0.45?
- It is equal to-- They would have to sell exactly 55.55
- glasses or 56 if I round.
- 55.5 repeating glasses.
- But you can't sell half a glass or we're assuming you
- can't sell half a glass.
- So the answer to that, they must sell 56 glasses.
- Because you can't sell half a glass, I'm assuming.
- So they need to sell 56 glasses to break even.
- Just to graph this.
- Once again, we'll hang out in the first quadrant because
- everything is going to be positive.
- Every glass they make $0.45.
- Let's say that they sell-- So this is the
- number of glasses, x.
- This is how much they make.
- Let me go by increments of 5.
- 5, 10, 15, 20, 25.
- Actually I need to go by even larger increments to get to
- the point that we're talking about.
- Let me go increments of 10.
- 10, 20, 30, 40, 50, 60.
- So that's the number of glasses.
- When they sell 0 glasses, they make $0.
- That's their y-intercept.
- y is equal to 0.
- When they sell 10 glasses, they make $4.50.
- So this is 4.50.
- This is 9.
- Actually let me just do it like this.
- Let me just mark only the multiples of 9.
- Let's say 9, 18, 27, 35.
- When they sell 10 glasses, they're going to make $4.50.
- 10 times 0.45.
- That's right there.
- 20 glasses, they're going to make $9.00.
- We can keep going there.
- 40 glasses, they're going to make $18.
- You see their slope.
- When you move 10, you're going to have to go up 4.50.
- This graph is going to look something like that.
- Should be a straight line.
- Then if you want to see their break even, their break even
- has to be $25, which is right about here.
- Their break even is $25 right around there.
- Let me draw the line a little bit better than that.
- The line is going to look like this.
- If the break even is $25-- it would be right there --you see
- that they have to sell about 56 glasses.
- Obviously the way I drew this isn't the super neatly drawn
- graph, but hopefully it gives you the general idea.