Math Help Needed

[KatieO]

We are stuck on problem and hope someone can help.

 

A bus and its driver can be rented for $350/day. The capacity for each bus is 60 students. Find an expression for the cost, c, of renting the buses needed for a trip as a function of n, the number of students going.

 

Thanks.

[Susan C.]

I sort of know, its been awhile since we did Algebra 2 over here. But I'm curious to how close I am....

 

I think you need to do something like c - 59 times $350 because if you have 61 students, you need 2 buses.

[AngieW in Texas]

You do need to do something so that you end up with whole buses.

 

C=350(n/60) would give you the answer if you could hire partial buses. What you need is an integer function that will round the answer to n/60 up to the next integer.

 

I can't think of what equation you'd have to have to figure out the number of buses aside from n/60 and then round up to the next integer. I'm pretty sure this is supposed to be an easy problem, but I can't come up with the answer right now. I'll think on it.

[forty-two]

A bus and its driver can be rented for $350/day. The capacity for each bus is 60 students. Find an expression for the cost, c, of renting the buses needed for a trip as a function of n, the number of students going.

The cost of the trip is given by g(n), where n is the number of students going:

g(n) = 350 * f(n)

f(n) =

/ n/60, if n modulo 60 == 0;

\ [n - (n modulo 60)]/60 + 1, if n modulo 60 != 0;

 

I think most Alg 2 programs use function notation; anyway, it was the easiest way to write it. You could just use one function, but it seemed clearer this way. There are more elegant ways to do it, but they need symbols I don't know how to type :tongue_smilie:.

[KatieO]

I really appreciate all your help but I'm still lost on this one. Can anyone simplify what I have to do. This has been driving me crazy all day.

[Dana]

We are stuck on problem and hope someone can help.

 

A bus and its driver can be rented for $350/day. The capacity for each bus is 60 students. Find an expression for the cost, c, of renting the buses needed for a trip as a function of n, the number of students going.

 

Thanks.

 

I'll give this a shot...

 

Total cost = (cost per bus)(# buses)

Total cost = 350(# buses)

 

That's the easy part :)

 

Then, the number of buses depends on the number of students.

Say we have 25 students. We need 1 bus.

Say we have 55 students. We still only need 1 bus.

 

Say we have 88 students. We need 2 buses.

130 students and we're up to 3 buses.

 

Back to students on the bus.

input (students) output (buses)

25 1

55 1

88 2

130 3

n still not really easy here... let's play around and see how we can use our earlier numbers and get a formula.

 

The number of buses needed depends on the number of students. Every 60 students gets 1 bus.

So 60 students = 1 bus,

120 students = 2 buses

180 students = 3 buses.

 

If we've got a number of students that's divisible by 60, we're fine - the number of buses is (students) / 60.

60 students gives 60/60 = 1 bus

120 students gives 120/60 = 2 buses

180 students gives 180/60 = 3 buses.

 

So what do we do when the number is not evenly divisible by 60?

 

25 students gives 25/60 ~ 0.42

55 students gives 55/60 ~ 0.92

88 students gives 88/60 ~ 1.47

130 students gives 130/60 ~ 2.17

 

But we can't drive around in a fractional bus :glare:

So we need a way to have only natural numbers for the number of buses.

Depending on what tools you have at this point in the text, you might approach this in different ways. The way I'll mention is using the greatest integer function (sometimes represented as brackets [ with double bars, for the internet, I'll just use brackets.

 

The greatest integer function is the greatest number that is less than or equal to the number that's input. So [1] = 1 but [0.5] = 0.

[0.42] = 0

[0.92] = 0

[1.47] = 1

[2.17] = 2

Hey... this is pretty close to the number of buses we need... except it's one less, so we want to add 1 to our answer.

 

students buses

25 [25/60] + 1 = 0+1 = 1

55 [55/60] + 1 = 0 + 1 = 1

88 [88/60] + 1 = 1 + 1 = 2

130 [130/60] + 1 = 2 + 1 = 3

Starting to look better!

n [n/60] + 1

 

Unfortunately, we do have a problem... remember the easy ones? If we had 180 students? Using this formula, that would give us [180/60]+1 buses or [3]+1 = 3+1 = 4 buses :(

There's probably a way around this to get a more elegant formula, but it's late here and I don't have time to play with it tonight.

 

So I'd state the answer in two ways...

Cost = 350(# buses)

 

If 60 is a factor of n, C(n) = 350(n/60)

If 60 is not a factor of n, C(n) = 350* ([n/60]+1)

 

I don't know if this is a problem you have the answer to and are trying to see where it comes from, but if you can give some more info (do you know about the greatest integer function at this point? are you using modulus arithmetic as was mentioned above?) then I may be able to play with it some more tomorrow... Let me know a bit more about the tools/techniques you have. I don't like it as a piecewise function, but that's all I've got for now!

 

Hope it helps some...

I find using words helps a lot and so does setting up charts for what you know is going on, then see if you can come up with a formula that matches the data.

[KatieO]

Dana, I REALLY am grateful for all the time you put into this. Unfortunately, I don't have the answer; its an online course (under the discrete function chapter) and the kids have not done the greatest integer function nor are they using modulus. Perhaps,it's just a bad question and the teacher should be queried about it. I had no problem with the equation for times when the passengers were a multiple of 60 but the other cases have me stumped completely. Thanks again!

[AngieW in Texas]

I did figure out a single formula that will work with the greatest integer function.

C=350(1+[(n-1)/60])

 

1+[(n-1)/60] will return the value you need.

for n=59, 1+[58/60]=1+0=1

for n=60, 1+[59/60]=1+0=1

for n=61, 1+[60/60]=1+1=2

 

However, you said that they haven't done the greatest integer function, so I'm not sure how they are supposed to come up with a formula.

[Dana]

I did figure out a single formula that will work with the greatest integer function.

C=350(1+[(n-1)/60])

 

 

I was thinking it was an n-1 somewhere but just wasn't seeing it! That'll keep this from bugging me during the day tomorrow. Thanks!!

 

Katie - good luck with it! I'd be interested in hearing the answer when you finally get it if it's a method that wasn't mentioned. (And I really should get to bed now!)

[KatieO]

Thanks all! Angie, even though we haven't done greatest integer functions, I'm wondering how you figured out the (n-1) part to your formula. Is there some special way to figure this type of thing out?

[AngieW in Texas]

No special trick. It's just that 1+[n/60] would give you an answer that was one too large for multiples of 60. If you subtract one from n, your answers will all come out right. I know I've had it pop up in problems before. I like math puzzles.

[KatieO]

Thanks!