How to find maximum change in the following scenario?

[PatternSeeker]

I need directions regarding methods that I could use for the following type of problem:

I am given the following scenario:

Observers consistently estimate objects as 20% shorter than they really are in the "y" dimension. They accurately estimate objects in the "x" dimension.

** error in estimated lengths in the y dimension, % error y = -20 % of physical lengths.
** error in estimated lengths in the x-dimension, % error x = 0% of physical length

Observers also consistently overestimate physical angles between "x" dimension and directions between "x" and "y" dimension.

Let physical angle = σ
Let estimated angle = β

** angle β = arctan ((sin α (% error x + 100))/(cos α ( % error y + 100))

*** Angles α and β vary between 0 deg to 90 degrees. The change between the
two is not constant, however. It will be greatest at a particular value of angle α.


QUESTION: If the physical length y is underestimated by 20 %, at which physical angle α
(between 0 and 90 degrees) will the change between angle β and angle α be
the greatest?

How would you suggest I approach this problem? Should I use differential calculus?

By the way, my background in math is pretty basic - I took undergraduate calculus a few years ago.

 

[tnich]

I need directions regarding methods that I could use for the following type of problem:

I am given the following scenario:

Observers consistently estimate objects as 20% shorter than they really are in the "y" dimension. They accurately estimate objects in the "x" dimension.

** error in estimated lengths in the y dimension, % error y = -20 % of physical lengths.
** error in estimated lengths in the x-dimension, % error x = 0% of physical length

Observers also consistently overestimate physical angles between "x" dimension and directions between "x" and "y" dimension.

Let physical angle = σ
Let estimated angle = β

** angle β = arctan ((sin α (% error x + 100))/(cos α ( % error y + 100))

*** Angles α and β vary between 0 deg to 90 degrees. The change between the
two is not constant, however. It will be greatest at a particular value of angle α.QUESTION: If the physical length y is underestimated by 20 %, at which physical angle α
(between 0 and 90 degrees) will the change between angle β and angle α be
the greatest?

How would you suggest I approach this problem? Should I use differential calculus?

By the way, my background in math is pretty basic - I took undergraduate calculus a few years ago.

Yes, differential calculus is the right approach to take. If it has been a while since you used it, you may need a reminder that the maximum or minimum of a differentiable function is at an end point of the function domain (in this case ##\alpha \in [0, 90\deg]##) or at a point where its derivative is zero (stationary point). So you will need to write an expression for the difference between ##\beta## and ##\alpha## and differentiate it to find a stationary point, and then check to see if it is a maximum or a minimum (or neither).