Distance Formula

[Colacanth]

What is the smallest possible value of the expression \(\displaystyle \sqrt{a^2+9}+\sqrt{(b-a)^2+4}+\sqrt{(8-b)^2+16}\) for real numbers \(\displaystyle a\) and \(\displaystyle b\)?
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This is from the section on the distance formula in my book, so it should somehow help with this problem.

Using the formula, the first, second and third terms are equal to the distance to \(\displaystyle (a,\,3)\), \(\displaystyle (b-a,\,2)\), and \(\displaystyle (8-b,\,4)\) from the origin \(\displaystyle (0,\,0)\) respectively.

However, this is where I get stuck and I have no idea where to progress from here.

Edit: Further thought shows that \(\displaystyle (0,\,0) \to (a,\,3)\), \(\displaystyle (a,\,0) \to (b,\,2)\), and \(\displaystyle (b,\,0) \to (8,\,4)\) work too.

 

[Colacanth]

Well, all 3 expressions are positive, right?

Yup.

Edit:

Further thought shows that \(\displaystyle (0,\,0) \to (a,\,3)\), \(\displaystyle (a,\,0) \to (b,\,2)\), and \(\displaystyle (b,\,0) \to (8,\,4)\) work too.

With that and the assumption that \(\displaystyle a < b < 8\), it seems that if I place the graph of each term where the previous term ended, I get a "slope" that spans 8 units on the x-axis and 9 units on the y-axis, and can be manipulated (with the x-values, since the y-values of each ordered pair are fixed) to form an actual slope of length \(\displaystyle \sqrt{145}\).

This would also make \(\displaystyle a = \dfrac{8}{3}\) and \(\displaystyle b = \dfrac{40}{9}\).

Edit\(\displaystyle _2\) (replying to the below posts): Thanks!

Edit\(\displaystyle _3\) All ​the below (and above) posts. And the ones on the next page too.

 

[Deleted member 4993]

Yup.

Edit:

With that and the assumption that \(\displaystyle a < b < 8\), it seems that if I place the graph of each term where the previous term ended, I get a "slope" that spans 8 units on the x-axis and 9 units on the y-axis, and can be manipulated (with the x-values, since the y-values of each ordered pair are fixed) to form an actual slope of length \(\displaystyle \sqrt{145}\).

This would also make \(\displaystyle a = \dfrac{8}{3}\) and \(\displaystyle b = \dfrac{40}{9}\).

Looks good to me .... Nice work

 

[Deleted member 4993]

Smallest integer solution: a=3 and b=5

With what constraint? Why not a=0 and b=0?

 

[lookagain]

.

Edit:

With that and the assumption that \(\displaystyle a < b < 8\), [/TEX].

At the outset of the problem:

I don't see assuming a < b. I don't see where there can be done "without a loss of generality."

I don't see discounting that a = b.

I don't see discounting b = 8.

 

[Deleted member 4993]

At the outset of the problem:

I don't see assuming a < b. I don't see where there can be done "without a loss of generality."

I don't see discounting that a = b.

I don't see discounting b = 8.

a=b or b = 8 does not give the minimum of the value of the total distance function.

 

[Novice]

Read the question carefully, it isn't asking the minimum values of a and b, but the minimum value of the expression. Carefully analysing, we see that the expression is the distance between
(0,0) to (a,3) + (a,3) to (b,1) + (b,1) to (8,-3). (Why convert to this form? You'll see later).
Now, we see that the path traced out by adding the three expressions goes from (0,0) to (8,-3). But the minimum distance between two points is the straight line joining them. Thus we see, the minimum value of the expression should be the distance between (0,0) and (8,-3) , which comes out to be approximately 8.5.
I leave the job of finding a and b up to you.
Hint: (a,3) and (b,1) lie on the line joining (0,0) and (8,-3). The slope of this line is the slope of the line joining (0,0) to (a,3), as well as the one joining (b,1) to (8,-3).

 

[Novice]

OK. But a=3 and b=5 is still minimum INTEGER solution...

Can't argue with that, except that that wan't asked, and the fact that this solution (integer solutions) can be found only by hit and trial, not by using the distance formula, which was the sole point of the question.

 

[Deleted member 4993]

OK. But a=3 and b=5 is still minimum INTEGER solution...

If by solution you mean answer to the given problem - [a=3, b=5] is not the solution. The problem wants the minimum value of the distance - not values of a & b.

- Cricket player

 

[Novice]

If by solution you mean answer to the given problem - [a=3, b=5] is not the solution. The problem wants the minimum value of the distance - not values of a & b.

- Cricket player

Even if the question asked the values of a and b, a=3, b=5 is not the correct solution.
The correct solution for values of a and b is (approximately), a= -8.05, b=-2.66.

 

[lookagain]

Even if the question asked the values of a and b, a=3,
b=5 is not the correct solution. The correct solution for values of a and b is
(approximately), a= -8.05, b=-2.66.

No, they are not (approximately) the values of a and b.

I don't even know where you got those negative signs from.

The values for a and b should both be positive, for one thing.



Edit:

Even (0,8) or (8, 8) provide shorter total distances than what you gave.

 

[Novice]

No, they are not (approximately) the values of a and b.

I don't even know where you got those negative signs from.

The values for a and b should both be positive, for one thing.



Edit:

Even (0,8) or (8, 8) provide shorter total distances than what you gave.

Thank you, lookagain. I actually did look again, and discovered the problem with my solution.

I had previously converted the expression to the form of distance between
(0,0) to (a,3) + (a,3) to (b,1) + (b,1) to (8,-3), and thought that this was the distance between (0,0) to (8,-3), but this was wrong, because we can clearly see that although the points (0,0), (a,3),(b,1) and (8,-3) lie in the same straight line, they do not lie on the line in this particular order, because of which we cannot say that adding up all these will give us the distance between (0,0) to (8,-3). Another way of seeing this is by noticing that the y coordinate is first increasing from 0 to 3, but then decreasing to 1, and decreasing again to -3, thus , the segments added up do not make the line, because we are going back and forth on the line while measuring distances of these segments.

I perhaps have gone too far in explaining it, and it might look complex, but it isn't.

To avoid this problem, I readjusted the expression to the form of distance between (0,0) to (8,-3) +(a,-3) to (b,-5) +(b,-5) to (8,-9),( I have done this so that a continuity is maintained among the y coordinates, and correspondingly, the x coordinates too) , from which the min value of expression(distance between (0,0) to (8,-9) came to be √145, as Colacanth had much before found out. The values of a and b are also the same as his, i.e.,
8/3, 40/9.