Making formulas and graphing them

[Kirito123]

Homework Statement

Individual tickets to the Mathtown Zoo cost $10 each. A Frequent Visit pass can be purchased for $60, and then each visit costs $5.

a. Write an equation that models the cost of individual tickets, C, as it relates to the number of tickets purchased, n.

b. Write an equation that models the cost of buying tickets with a Frequent Visit pass, C, as it relates to the number of tickets purchased, n.

c. Create a graph that shows both relations.

Homework Equations

The Attempt at a Solution

a. an equation that models the cost of individual tickets can be written as:

t = c x n
Where t is the total cost, c represents the individual tickets, and n represents number of tickets purchased.b. equation can be written as: t = cn + 60

Where c represents the cost of a ticket with a frequent pass ($5), n represents the number of tickets purchased, t represents the total cost and 60 represents the cost to buy the frequent visit pass.

c. This was my main problem, wouldn't the graph be huge since we have to show both relations of the 2 formulas? for example if my 2 formulas are correct how would i fit the "t=cn + 60" formula. it would be huge, for eg so you purchase one ticket and it cost $5. That there would have to be at the point 65.

If i made no sense above sorry .
anyway is the formulas wrong and if not would i have to make the graph large or just use larger numbers as in on the y-axis it would be like : 10, 20 ,30 etc?

 

[DrClaude]

a. an equation that models the cost of individual tickets can be written as:

t = c x n
Where t is the total cost, c represents the individual tickets, and n represents number of tickets purchased.b. equation can be written as: t = cn + 60

It is probably better to use numbers instead of c, especially since c is not the same in both formulas.

c. This was my main problem, wouldn't the graph be huge since we have to show both relations of the 2 formulas? for example if my 2 formulas are correct how would i fit the "t=cn + 60" formula. it would be huge, for eg so you purchase one ticket and it cost $5. That there would have to be at the point 65.

If i made no sense above sorry .
anyway is the formulas wrong and if not would i have to make the graph large or just use larger numbers as in on the y-axis it would be like : 10, 20 ,30 etc?

You of course have to use a proper scaling. For such a problem, I would probably take n to go from 1 to 20, and scale y appropriately such that all values fit in the graph.

 

[Kirito123]

No they told me in my homework to represent individual tickets as c for question 1 and for the next question c represents the number of tickets but with a frequent visit pass.

Ok ill take you advice about the graphing :)

 

[DrClaude]

No they told me in my homework to represent individual tickets as c for question 1 and for the next question c represents the number of tickets but with a frequent visit pass.

Ok, I now understand the problem statement better. But that means that your equations are not correct. What is asked for is
C = ...
where C is the price per ticket when buying n tickets.

 

[Kirito123]

Isn't it different for each questions. since for question a they told me to represent individual tickets as c and for question b i had to represent buying tickets with a Frequent Visit pass as c also.

In this case c has a different meaning for each question?

What is asked for is
C = ...

If they are asking for c, why are they wanting a formula to model these equations??

 

[DrClaude]

Isn't it different for each questions. since for question a they told me to represent individual tickets as c and for question b i had to represent buying tickets with a Frequent Visit pass as c also.

In this case c has a different meaning for each question?

Yes, you should get two different functions C(n).

If they are asking for c, why are they wanting a formula to model these equations??

Because they want the effective cost per ticket, which can depend on the number of tickets bought. In b, if you buy a single ticket, you will have paid $65. But if you buy 2 tickets, you will not have paid 2 × $65 = $130, so the cost per ticket is $65 when n = 1 but not when n = 2.

 

[Kirito123]

But if you buy 2 tickets, you will not have paid 2 × $65 = $130

Just asking i might be wrong why did you multiply $65 by 2 (tickets).

In my formula t = cn + 60. If i plugged in 2 tickets it would be:

5 x 2 = 10
10 + 60 would equal to 70.

so how come you multiplied the tickets by the cost of the frequent pass??
If I'm wrong I probably didn't quite get what you are trying to say (sorry).

 

[DrClaude]

Just asking i might be wrong why did you multiply $65 by 2 (tickets).

Because since C = $65 for n = 1, that's how much you would pay per ticket for two tickets if the cost C was independent of n.

In my formula t = cn + 60. If i plugged in 2 tickets it would be:

5 x 2 = 10
10 + 60 would equal to 70.

The problem asks for an equation for C, the cost per ticket, not the total cost.

If I'm wrong I probably didn't quite get what you are trying to say (sorry).

The problem is that I'm trying to explain without giving you the answer.

 

[Kirito123]

How does c = 65? didn't i say c is equal to buying tickets with a Frequent Visit pass. after you bought a visit pass the cost becomes $5. The 60 dollars is for buying the pass itself.

I think i may have found a error in my formula: cn+60. Since we only need to buy the visit pass once. so first time it would cost 65 the second time it should only cost $5 since there's not additional fee?

 

[MullaTheMech]

So with the formulas you've created you get a big jump of 60 and then small increments of 5.
In high school I got away with breaking a graph during initial jumps. I would put a break in the y-axis right at the bottom. My teachers never liked breaks in graphs but I could get away with breaks in the beginning of an axis. A break looks like sharp back and forth lines like a heartbeat on a heartbeat monitor.

 

[DrClaude]

How does c = 65? didn't i say c is equal to buying tickets with a Frequent Visit pass. after you bought a visit pass the cost becomes $5. The 60 dollars is for buying the pass itself.

Yes, but the exercise only makes sense if it is asking for the effective cost per ticket. Otherwise, you just get a straight line: C = $10 or C = $5. Whereas, the effective cost for 1 ticket with the Frequent Visit pass is $65, the cost for 2 tickets $35, etc. Graphing the two functions, you will see that they will cross at a certain point: the number of tickets at which the Frequent Visit pass becomes worth the investment.

 

[Kirito123]

you will see that they will cross at a certain point: the number of tickets at which the Frequent Visit pass becomes worth the investment.

when you say it becomes worth the investment, you mean when it become better to buy the frequent pass, as in how many times you should visit until it becomes best to buy. I did the equation below and I'm am pretty sure its right. If i made a mistake my bad.10n = 60 + 5n

10n – 5n = 60 (subtract 5n from both sides of the equations)

5n = 60 (divide both sides by 5)

N = 60/5

N = 12 (n is equal to 12)

If Sherry went 12 times to the zoo, she would have to pay $120, which is equivalent to the amount of money she would have to pay purchasing the regular tickets.

Equation for regular tickets:

t = c x n

t = $10 x 12 visits = $120

Equation when purchasing the frequent visit pass:

t = cn + 60

t = $5 x 12 visits + 60 = $120

As shown above, if you were to visit the zoo 12 times, regardless of the deal, she will still have to pay $120. This indicates that sherry will have to visit the zoo more than 12 times, in order to receive a better buy.

Equation for regular tickets:

t = c x n

t = $10 x 13 visits = $130

Equation when purchasing the frequent visit pass:

t = cn + 60

t = $5 x 13 visits + 60 = $125If sherry visited the zoo 13 times she would pay less with the frequent visit pass deal ($125) compared to what she would have to pay for regular tickets ($130). So as you can see above, sherry will start to save more money, if she visits the zoo more than 12 times.