Why Does One Mathematical Approach Fail in This Optimization Problem?

[ohlhauc1]

We had to do a question in my Advanced Mathematics class, and the way that I did the problem was supposedly right. My teacher even did it that way. However, the answer was wrong, so my teacher showed us the correct way to do the problem.

The dilemma I have is that BOTH ways should work, but they don't. Therefore, I am going to give both solutions to the problem, and I was wondering if you could tell me why the first one does not work.

Question: The coordinates of points P and Q are (1,2) and (2,-3), respectively, and R is a point on the line x=-1. Find the coordinates of R so that PR + RQ is a minimum?

Here is the graph/image I used for this problem:
http://s62.yousendit.com/d.aspx?id=0LUTJESKXM0R33IOBSY2IH1Z3A

MY SOLUTION:

Equations: PR^2 = 2^2 + y^2
QR^2 = 3^2 + (5 - y)^2

PR^2 + QR^2 = minimum or m
2^2 + y^2 + 3^2 + (5 -y)(5 - y) = minimum
4 + y^2 + 9 + 25 - 10y + y^2 = minimum
2y^2 - 10y + 38 = minimum

y = -b/2a = 10/4 = 5/2

Y-coordinate of R = 2 - 5/2 = -1/2

Therefore, the coordinates of R are (-1, -1/2)

CORRECT SOLUTION:

Let m = minimum

mP'R = [ (2 - y) / (-3 + 1) ] = [ (2 - y) / -2 ]

mRQ = [ (y + 3) / (-1 -2) ] = [ (y + 3) / -3 ]

Note: mP'R = mRQ
[ (2 - y) / -2 ] = [ (y + 3) / -3 ]
= -2y - 6 = 3y - 6
= -5y = 0
y = 0

The coordinates of R are therefore (-1, 0).

*So...I have presented both solutions, and I would highly appreciate your help. I am really interested in trying to understand this dilemma.

Thanks

 

[NateTG]

The maximum of the sum of the squares isn't necessarily the maximum of the sum.

 

[Architeuthis Dux]

for sharing your question and dilemma about this problem in your Advanced Mathematics class. I am happy to provide my response and help you understand this dilemma.

First of all, it is important to note that there can be multiple ways to solve a problem, and both of your solutions are valid approaches. However, the key to finding the correct solution is to carefully consider the given information and use the appropriate mathematical concepts and equations.

In your first solution, you used the distance formula to find the minimum value of PR^2 + QR^2. However, this approach assumes that the point R can be any point on the line x = -1, which is not true. The problem states that R is a point on the line x = -1, which means that its x-coordinate is always -1, and only the y-coordinate can vary. Therefore, your solution does not take into account the constraint that R must lie on the line x = -1.

In the correct solution, the concept of minimizing the distance between two points is used. This approach considers the given information that R must lie on the line x = -1, and uses the slope of the line to find the y-coordinate of R that minimizes the distance between P and R, and between R and Q. This approach takes into account the constraint of R being on the line x = -1 and gives the correct solution.

In conclusion, the key to finding the correct solution to this problem is to carefully consider the given information and use the appropriate mathematical concepts. Both of your solutions are valid approaches, but the second one takes into account the given constraint and gives the correct answer. I hope this helps you understand the dilemma and provides some clarity on why the first solution did not work. Keep up the good work in your Advanced Mathematics class!