What is being asked for here?

[allegansveritatem]

Here is the problem:


What is meant by "show"? Here is the only thing I could think of doing with part (a):


Y here is distance apart of cars and X is number of cars. Thus, if there were zero distance between cars then you could fit 440 cars on the bridge. Is that what "show" means here?
What about part (b)? Supposing part (a) is correct, then I could take the Y values and use them in an equation to get v. But I am not sure that is what the author of the problem is after.

 

[Dr.Peterson]

"Show" means to derive that formula.

What is the total distance covered by n cars, under the indicated conditions? Write an inequality that says they all fit on the bridge, and solve for n. Then find the largest integer value.

Note that their answer is technically a little off; you might come up with a slightly different answer, if you are more careful than they were. Don't let that bother you.

 

[Otis]

It helps to sketch some examples, perhaps experimenting with a much shorter bridge while considering gaps where d is both longer and shorter than one car. You'll realize the number of gaps with length d is not necessarily the same as the number of cars (i.e., possibilities exist where the last car on the bridge fits without a full gap of length d behind it -- or any gap at all, in some cases). The exercise author overlooked that fact, and that's why their expression for the maximum number of cars is off (for certain values of d).

For part (a), I also wrote the inequality suggested in post #2. If you'd like help finding n, please show how far you got.

?

 

[allegansveritatem]

"Show" means to derive that formula.

What is the total distance covered by n cars, under the indicated conditions? Write an inequality that says they all fit on the bridge, and solve for n. Then find the largest integer value.

Note that their answer is technically a little off; you might come up with a slightly different answer, if you are more careful than they were. Don't let that bother you.

Write an inequality...I will have to work that out today. Right now I have no idea what it will look like. I will get back tonight and post my results. Thanks for tip.

 

[allegansveritatem]

It helps to sketch some examples, perhaps experimenting with a much shorter bridge while considering gaps where d is both longer and shorter than one car. You'll realize the number of gaps with length d is not necessarily the same as the number of cars (i.e., possibilities exist where the last car on the bridge fits without a full gap of length d behind it -- or any gap at all, in some cases). The exercise author overlooked that fact, and that's why their expression for the maximum number of cars is off (for certain values of d).

For part (a), I also wrote the inequality suggested in post #2. If you'd like help finding n, please show how far you got.

?

I am going to give it another go today. I will post my results here tonight. I will certainly make a scribble sketch of the situation...that usually helps. I guess the author didn't include all the nuances of the...is it gestalt? because he didn't want to contribute to the bewilderment of students of my stripe any more than was necessary. Thanks for working it out. I will see how close I can come today to a happy conclusion of this dilemma.

 

[allegansveritatem]

I did what I could with this problem along the lines advised above. Here are my results for parts a and b:




I confess I didn't quite know which way to face the sign. these two results seem to make sense--I fooled with both of the problems switching around the variables...but really wasn't on solid ground. Is this anywhere near a acceptable answer to the problem?

 

[Dr.Peterson]

Showing that something is true does not mean to start with what they claim is true! You can't start with something you don't know to be true.

Start with what Otis said:

It helps to sketch some examples, perhaps experimenting with a much shorter bridge while considering gaps where d is both longer and shorter than one car.

In fact, sketch what it looks like if just 2 cars are on the bridge; then 3 cars. How far will they extend?

Then do what I suggested, writing an expression for the total distance of n cars:

What is the total distance covered by n cars, under the indicated conditions? Write an inequality that says they all fit on the bridge, and solve for n. Then find the largest integer value.

Do that, and then we can talk about what to do next.

 

[allegansveritatem]

Showing that something is true does not mean to start with what they claim is true! You can't start with something you don't know to be true.

Start with what Otis said:

In fact, sketch what it looks like if just 2 cars are on the bridge; then 3 cars. How far will they extend?

Then do what I suggested, writing an expression for the total distance of n cars:

Do that, and then we can talk about what to do next.

OK. I'll give it another try.

 

[allegansveritatem]

So, I went at this again today--and yesterday too for that matter--but could not come to much in the way of setting up inequalities. Finally I just went through a process of reasoning and here is what I came up with:
Part a)


Part b)


I don't think this is exactly what the author had in mind but....

 

[Deleted member 4993]

I would have started with a "mental" sketch of a different problem (simpler but similar).

Suppose the cars do not have any space in front or in rear

How many cars can you fit in a 24' long bridge?

How many cars can you fit in a 36' long bridge?

How many cars can you fit in a 40' long bridge?

Now start to think about the space. Assign half the space (d/2) in front and half the space in rear

At this point, real sketch will start to help.

Continue.....

 

[Dr.Peterson]

So, I went at this again today--and yesterday too for that matter--but could not come to much in the way of setting up inequalities. Finally I just went through a process of reasoning and here is what I came up with:

I don't think this is exactly what the author had in mind but....

I don't know what the author expected; this is not what I was leading you toward, but your work for part (a) is not inappropriate, and you do seem to have the right picture in mind. (I think my suggestion is better for a more complicated problem, and therefore worth learning to do.) The tricky part in your approach is perhaps in deciding for sure which direction to round. Your answer isn't quite right in that regard.

Here is the inequality approach I had in mind:

Suppose there are n cars on the bridge. Then if each one takes up 12 + d feet, then all together they take up n(12 + d) feet, right? (Otis and I both alluded to a slight inaccuracy here, but I'll let it go, since this will lead to their answer.​

​

Now, if they all fit on the bridge, this distance must be no more than the length of the bridge:​

​

n(12 + d) ≤ 5280​

​

Solving for n, we get​

​

n ≤ 5280/(12 + d)​

​

Now, what is the largest number of cars that can fit? It's the largest integer that is no greater than that RHS. That's what the greatest integer function means:​

​

n = [[5280/(12 + d)]]​

(Your notation isn't what they intended. It's not a double absolute value, like ||x||, but a double bracket, like [[x]], which denotes, as they say, the "greatest integer function" -- that is the greatest integer that is not greater than x. It means, that is, to round down (not "to the greatest integer", which sounds like it means rounding up. Some people just use [x], though that can be misleading; others use floor(x) or [MATH]\lfloor x \rfloor[/MATH] with the same meaning.)

As I suggested, using an inequality makes it a little easier to be sure how to round, though I imagine you actually meant the right thing.

Now see if you can make your answer to part (b) a little more convincing. Use the letter F and v throughout, rather than using a specific number but popping F in there magically at the end and never getting back to v at all. The right ideas are there, but a proof has to prove, not just suggest. (Yes, their word is "show", but you have to really show.)

 

[allegansveritatem]

I would have started with a "mental" sketch of a different problem (simpler but similar).

Suppose the cars do not have any space in front or in rear

How many cars can you fit in a 24' long bridge?

How many cars can you fit in a 36' long bridge?

How many cars can you fit in a 40' long bridge?

Now start to think about the space. Assign half the space (d/2) in front and half the space in rear

At this point, real sketch will start to help.

Continue.....

I will have to think about this...I think this is one of the ways I was considering but I considered so many ways that I'm not sure. Thanks for the suggestion. I will ponder it.

 

[allegansveritatem]

I don't know what the author expected; this is not what I was leading you toward, but your work for part (a) is not inappropriate, and you do seem to have the right picture in mind. (I think my suggestion is better for a more complicated problem, and therefore worth learning to do.) The tricky part in your approach is perhaps in deciding for sure which direction to round. Your answer isn't quite right in that regard.

Here is the inequality approach I had in mind:

Suppose there are n cars on the bridge. Then if each one takes up 12 + d feet, then all together they take up n(12 + d) feet, right? (Otis and I both alluded to a slight inaccuracy here, but I'll let it go, since this will lead to their answer.​

​

Now, if they all fit on the bridge, this distance must be no more than the length of the bridge:​

​

n(12 + d) ≤ 5280​

​

Solving for n, we get​

​

n ≤ 5280/(12 + d)​

​

Now, what is the largest number of cars that can fit? It's the largest integer that is no greater than that RHS. That's what the greatest integer function means:​

​

n = [[5280/(12 + d)]]​

(Your notation isn't what they intended. It's not a double absolute value, like ||x||, but a double bracket, like [[x]], which denotes, as they say, the "greatest integer function" -- that is the greatest integer that is not greater than x. It means, that is, to round down (not "to the greatest integer", which sounds like it means rounding up. Some people just use [x], though that can be misleading; others use floor(x) or [MATH]\lfloor x \rfloor[/MATH] with the same meaning.)

As I suggested, using an inequality makes it a little easier to be sure how to round, though I imagine you actually meant the right thing.

Now see if you can make your answer to part (b) a little more convincing. Use the letter F and v throughout, rather than using a specific number but popping F in there magically at the end and never getting back to v at all. The right ideas are there, but a proof has to prove, not just suggest. (Yes, their word is "show", but you have to really show.)

Yes, I see it. Very economical. In one of my attempts I did do something like this: I assigned a value to d of 10 ft and divided 5280 by 22. I got 240 as N and ...I'm not sure where I went from there. I will overhaul my b) proof. I just read it over. Pretty sloppy. As for the greatest integer...yes, I knew that it meant to round down but I didn't know that particular phrase--a good one. I must say that thinking about this situation for as long as I did had a certain value. If I were to spend that much time with each of the word problems--or rather, each of the types of word problem--I would probably be able to do individual incarnations of said types with a lot more confidence and speed. This is something I may try, at least in a small way. I will have another go at part b) and post it. Thanks very much for the pointers.

 

[gandrew]

I don't know what the author expected; this is not what I was leading you toward, but your work for part (a) is not inappropriate, and you do seem to have the right picture in mind. (I think my suggestion is better for a more complicated problem, and therefore worth learning to do.) The tricky part in your approach is perhaps in deciding for sure which direction to round. Your answer isn't quite right in that regard.

Here is the inequality approach I had in mind:

Suppose there are n cars on the bridge. Then if each one takes up 12 + d feet, then all together they take up n(12 + d) feet, right? (Otis and I both alluded to a slight inaccuracy here, but I'll let it go, since this will lead to their answer.​

​

Now, if they all fit on the bridge, this distance must be no more than the length of the bridge:​

​

n(12 + d) ≤ 5280​

​

Solving for n, we get​

​

n ≤ 5280/(12 + d)​

​

Now, what is the largest number of cars that can fit? It's the largest integer that is no greater than that RHS. That's what the greatest integer function means:​

​

n = [[5280/(12 + d)]]​

(Your notation isn't what they intended. It's not a double absolute value, like ||x||, but a double bracket, like [[x]], which denotes, as they say, the "greatest integer function" -- that is the greatest integer that is not greater than x. It means, that is, to round down (not "to the greatest integer", which sounds like it means rounding up. Some people just use [x], though that can be mcdvoice misleading; others use floor(x) or [MATH]\lfloor x \rfloor[/MATH] with the same meaning.)

As I suggested, using an inequality makes it a little easier to be sure how to round, though I imagine you actually meant the right thing.

Now see if you can make your answer to part (b) a little more convincing. Use the letter F and v throughout, rather than using a specific number but popping F in there magically at the end and never getting back to v at all. The right ideas are there, but a proof has to prove, not just suggest. (Yes, their word is "show", but you have to really show.)

i was solving it and got stuck in some steps but this helped me to it out thanks for taking your time to explain