too little info?

[allegansveritatem]

Here is problem:


here is the accompanying image:


Here is what I did with a)


This seems right but... for whaterver input of s I use, I keep getting the same d. One. What gives? I am thinking that I need more info to get the constant. No?

 

[MarkFL]

We are told to let:

[MATH]L=s=10[/MATH]
And so we then have:

[MATH]d=2000c[/MATH]
Now, letting \(d=1\), we can find \(c\):

[MATH]1=2000c\implies c=\frac{1}{2000}[/MATH]
Now for part b), we should begin with:

[MATH]\frac{1}{2}=\frac{1}{2000}s^2(30-s)[/MATH]
[MATH]s^3-30s^2+1000=0[/MATH]
Using a numeric root finding technique, we find the root satisfying the given conditions to be:

[MATH]s\approx6.52703644666139[/MATH]

 

[Dr.Peterson]

This seems right but... for whatever input of s I use, I keep getting the same d. One. What gives? I am thinking that I need more info to get the constant. No?

You appear to have missed the fact that the deflection at the end of the board is 1 foot, so s = L for this calculation. From that you can find c. (After finding the constant, you can "set s free again", since c is a constant and only has to be calculated once.)

For part (b), it will be sufficient to put s = 6.5 and s = 6.6 into the formula, and show that these values imply that d = 1/2 somewhere in that interval. You may have been taught a theorem that implies this, or you may just be expected to use common sense.

 

[allegansveritatem]

We are told to let:

[MATH]L=s=10[/MATH]
And so we then have:

[MATH]d=2000c[/MATH]
Now, letting \(d=1\), we can find \(c\):

[MATH]1=2000c\implies c=\frac{1}{2000}[/MATH]
Now for part b), we should begin with:

[MATH]\frac{1}{2}=\frac{1}{2000}s^2(30-s)[/MATH]
[MATH]s^3-30s^2+1000=0[/MATH]
Using a numeric root finding technique, we find the root satisfying the given conditions to be:

[MATH]s\approx6.52703644666139[/MATH]

I see what you are doing but I am still not clear on how you get L and s to both equal 10. Maybe I'm not understanding what s means.

 

[allegansveritatem]

You appear to have missed the fact that the deflection at the end of the board is 1 foot, so s = L for this calculation. From that you can find c. (After finding the constant, you can "set s free again", since c is a constant and only has to be calculated once.)

For part (b), it will be sufficient to put s = 6.5 and s = 6.6 into the formula, and show that these values imply that d = 1/2 somewhere in that interval. You may have been taught a theorem that implies this, or you may just be expected to use common sense.

But I don't see how s and L are both 10. How do you come by that? Maybe I am being influenced by the diagram, which clearly shows s NOT to be as long as the board. Does s mean "the place where the diver is standing" as the time of the measurement? In which case I can see how s and L are equivalent. Otherwise.... s is clearly not L.

 

[MarkFL]

I see what you are doing but I am still not clear on how you get L and s to both equal 10. Maybe I'm not understanding what s means.

We are told for part a) that the deflection is at the end of the board, where \(s=L\), and we are given \(L=10\text{ ft}\).

 

[Dr.Peterson]

d

But I don't see how s and L are both 10. How do you come by that? Maybe I am being influenced by the diagram, which clearly shows s NOT to be as long as the board. Does s mean "the place where the diver is standing" as the time of the measurement? In which case I can see how s and L are equivalent. Otherwise.... s is clearly not L.

Read the problem carefully:

The deflection d of the board at a position s feet from the stationary end is given by [MATH]d = c s^2(3L 0 s)[/MATH] for [MATH]0 \le s \le L[/MATH], where L is the length of the board and c is a positive constant ... Suppose the board is 10 feet long.​

​

(a) If the deflection at the end of the board is 1 foot, find c.​

So s is the place where we are measuring deflection, which can be anywhere from 0 (at the stationary end) to L (at the place where the diver is).

For the entire problem, we are told that L = 10.

For the scenario of (a), we are told that when s = L (that is, we are measuring deflection at the end of the board), d = 1.

So we put L = 10, s = 10, and d = 1 into the formula and solve for c.

 

[allegansveritatem]

d
Read the problem carefully:

The deflection d of the board at a position s feet from the stationary end is given by [MATH]d = c s^2(3L 0 s)[/MATH] for [MATH]0 \le s \le L[/MATH], where L is the length of the board and c is a positive constant ... Suppose the board is 10 feet long.​

​

(a) If the deflection at the end of the board is 1 foot, find c.​

So s is the place where we are measuring deflection, which can be anywhere from 0 (at the stationary end) to L (at the place where the diver is).

For the entire problem, we are told that L = 10.

For the scenario of (a), we are told that when s = L (that is, we are measuring deflection at the end of the board), d = 1.

So we put L = 10, s = 10, and d = 1 into the formula and solve for c.

OK. I think I have it now. We are told that the deflection is 1 foot and that the deflection is at the end of the board. Baked into the deflection, so to speak, is s--s being the distance from the root of the board (as it were) to the deflection point. And for a) we have been told that the deflection point is at the end of the board and we have also been (kindly) told that the board is ten feet long. The rest, as the feller said, is a piece of cake. My problem is inattentive reading. Thanks for pointing these things out.
As an aside, it says in the description that the constant, c, is dependent on (or a function of) the weight of the diver and the physical properties of the board--by which I suppose is meant the material the board is made of and how flexible said material is. But...it is also a function of s, no? I mean if the formula given is true then c=d/s^2(3L-s), no? So everything is tangled up with everything else...and as I think about it, that is the way with every constant. No?

 

[allegansveritatem]

We are told for part a) that the deflection is at the end of the board, where \(s=L\), and we are given \(L=10\text{ ft}\).

I get it now. Thanks

 

[Dr.Peterson]

As an aside, it says in the description that the constant, c, is dependent on (or a function of) the weight of the diver and the physical properties of the board--by which I suppose is meant the material the board is made of and how flexible said material is. But...it is also a function of s, no? I mean if the formula given is true then c=d/s^2(3L-s), no? So everything is tangled up with everything else...and as I think about it, that is the way with every constant. No?

Constant means constant! They're saying this is a fixed number for all situations under consideration (because it is always the same board and same diver), and independent of the variables in the formula.

Yes, you can calculate c from the variables using the formula you mention (as in fact you did for part a), but the fact that it is a constant means that you will always get the same number. The variables can't have arbitrary values.

 

[allegansveritatem]

Constant means constant! They're saying this is a fixed number for all situations under consideration (because it is always the same board and same diver), and independent of the variables in the formula.

Yes, you can calculate c from the variables using the formula you mention (as in fact you did for part a), but the fact that it is a constant means that you will always get the same number. The variables can't have arbitrary values.

OK. I will have to ponder this.

 

[allegansveritatem]

OK. I will have to ponder this.

what I meant by the constant being a function of everything in the situation or gestalt that we call "the problem" is, to use the present problem as an example, that the constant depends on the diver's weight and the composition of the board..but the diver's weight is what deflects the board and thus the deflection is implicated in the make-up of the constant as much as the other two factors. All of which doesn't mean that it changes casually or randomly....well, anyway, I think I get it about the constancy of the constant.