how to maximize the value of z?

[PG1995]

Hi,

When a distance of 1 unit is moved along x-axis, there is 1 unit height increase along z-axis. Likewise, when a distance of 1 unit is moved along y-axis, there is 2 unit height increase along z-axis. It means that when you move a distance of √2 along diagonal, there would be a 3-unit height increase along z-axis. Please see the the picture.

We only want to move 1-unit along the diagonal and still want the height along z-axis to be maximum. It means that we need a certain combination of x- and y-units.

I think that 0.45 units along x-axis and 0.9 units along y-axis will give us the maximum height along z which is 2.25. Please have a look here.

I'm not sure if the combination given above is correct, and also how the method works. It looks like an optimization problem. Not sure. Could you please guide me? Thank you.

 

[Ratchit]

Hi,

When a distance of 1 unit is moved along x-axis, there is 1 unit height increase along z-axis. Likewise, when a distance of 1 unit is moved along y-axis, there is 2 unit height increase along z-axis. It means that when you move a distance of √2 along diagonal, there would be a 3-unit height increase along z-axis. Please see the the picture.

We only want to move 1-unit along the diagonal and still want the height along z-axis to be maximum. It means that we need a certain combination of x- and y-units.

I think that 0.45 units along x-axis and 0.9 units along y-axis will give us the maximum height along z which is 2.25. Please have a look here.

I'm not sure if the combination given above is correct, and also how the method works. It looks like an optimization problem. Not sure. Could you please guide me? Thank you.

Your answer is correct. X-distance is 1/sqrt(5 )= 0.447214, and y-distance is 2/sqrt(5) = 0.894427 . Did you solve it by incremental iteration or by differential calculus? It is a simple calculus problem.

Ratch

 

[PG1995]

Thank you, Ratch.

I was discussing it with someone and I was given the number but even that person wasn't really sure about it. I don't know how he came up with these numbers. How do you do it by differential calculus? I'd prefer to know about differential calculus method than iteration method which is more of a 'mechanical' algorithm. Could you please help? Thanks.

 

[Ratchit]

Thank you, Ratch.

I was discussing it with someone and I was given the number but even that person wasn't really sure about it. I don't know how he came up with these numbers. How do you do it by differential calculus? I'd prefer to know about differential calculus method than iteration method which is more of a 'mechanical' algorithm. Could you please help? Thanks.

If the x-distance is "x", what is the y-distance?

Ratch

 

[PG1995]

If the x-distance is "x", what is the y-distance?

"y"! I'm sorry that I don't get your point.

 

[Ratchit]

"y"! I'm sorry that I don't get your point.

The diagonal is a function of the x-distance and the y-distance. If you know the x-distance, you can easily calculate the y-distance. Hint: Pythagoras's theorem

Ratch

 

[PG1995]

It looks like that you have misinterpreted the original query.

We don't know the x-distance.

The problem is about x,y,z plane. We are given that when a distance of 1 unit is moved along x-axis, there is 1 unit height increase along z-axis. Likewise, when a distance of 1 unit is moved along y-axis, there is 2 unit height increase along z-axis. It means that when you move a distance of √2 along diagonal, there would be a 3-unit height increase along z-axis.

Then, using the given slope ratios, i.e. 1-z/1-x and 2-z/1-y, we are asked to find the combination of optimum x-distance and y-distance which gives maximum height for z.

Thank you.

 

[Ratchit]

It looks like that you have misinterpreted the original query.

We don't know the x-distance.

The problem is about x,y,z plane. We are given that when a distance of 1 unit is moved along x-axis, there is 1 unit height increase along z-axis. Likewise, when a distance of 1 unit is moved along y-axis, there is 2 unit height increase along z-axis. It means that when you move a distance of √2 along diagonal, there would be a 3-unit height increase along z-axis.

Then, using the given slope ratios, i.e. 1-z/1-x and 2-z/1-y, we are asked to find the combination of optimum x-distance and y-distance which gives maximum height for z.

Thank you.

I fully understand the problem. You have not answered the question I asked. If I move x amount of distance along the x-axis until the diagonal is one, how much will I move along the y-axis. It is a simple geometric relationship. Use the "hint" I gave you previously. Forget about the "z" direction for now.

Ratch

 

[PG1995]

Thanks.

Pythagoras theorem:
hypotenuse^2 = base^2 + perpendicular^2
diagonal^2 = x-distance^2 + y-distance^2
y-distance^2 = diagonal^2 - x-distance^2
y-distance = sqrt{diagonal^2 - x-distance^2}
y-distance = sqrt{1 - x-distance^2}

I hope that this is what you were asking for.

 

[Ratchit]

Thanks.

Pythagoras theorem:
hypotenuse^2 = base^2 + perpendicular^2
diagonal^2 = x-distance^2 + y-distance^2
y-distance^2 = diagonal^2 - x-distance^2
y-distance = sqrt{diagonal^2 - x-distance^2}
y-distance = sqrt{1 - x-distance^2}

I hope that this is what you were asking for.

Correct. So now you know the y-distance in terms of the x-distance. So we can write the height equation. It is:

So that equation determines height Z if the diagonal is one and x is known. Now, using your knowledge of differential calculus, can you determine the value of x such that Z is at its maximum? And, once you know x, you can easily calculate the y-distance.

Ratch

 

[PG1995]

Thanks.

Is it correct?

y-distance = sqrt{1 - x-distance^2}=sqrt{1 - 0.45^2}=0.9

 

[Ratchit]

Thanks.

Is it correct?

y-distance = sqrt{1 - x-distance^2}=sqrt{1 - 0.45^2}=0.9

From what I can see of the calculations, it appears to be correct. Much of the image is truncated, however.

Ratch

 

[PG1995]

From what I can see of the calculations, it appears to be correct. Much of the image is truncated, however.

There are only two lines of calculations. First, derivative is found and then maximum value of z is found by setting the found derivative expression to zero.

It looks like that it's settled. Thank you so much!