Question

George is currently located at the point (4, 3). He has been invited to his friend's house for dinner at (7, 8), but said he would bring dessert, and so he needs to visit the local supermarket 'chain' which has stores all along the x-axis. What is the least number of units that George must walk to get to the store, and then to his friends house (ignore time inside store)?

Answer 1

What part of this problem is giving you trouble?

Answer 2

Well, I can't seem to figure out the 'shortest distance' part. I could just find the distance perpendicular to the x-axis from one of the coordinates, but I doubt that it would be the shortest distance.

Answer 3

Have you drawn a diagram?

Answer 4

No, but I can clearly visualize it. I just can't figure out how to draw the path.

Answer 5

I would strongly suggest sketching it out. The two points, and an arbitrary point on the x axis somewhere between 4 and 7. Next step would be to find the two distances -- house to x axis, and x axis to friend's house.

Answer 6

Well, the answers they give rarely sketch it out and they generally want you to just use algebra to figure it out...The answer is \(\sqrt{130}\) is this just a guess and check method by plugging in numbers within the range 4 < x < 7?

Answer 7

There is a calculus way of solving this. Do you remember the distance formula?

Answer 8

Yes.
\[d = \sqrt{(x_{1} - x_{2})^{2} + (y_{1} - y_{2})^{2}}\]

Answer 9

Ok. In this problem you want to minimize the SUM of the distances between A (home) , C (the x-axis somewhere between 4 and 7) and B (your friend's house).

Answer 10

Ok, but that's where I'm stuck at. The only way I can see of solving this is using the distance formula to guess and check.

Answer 11

The sum of the distances will ultimately be a function of x. To find its minima / maxima, use the 1st derivative test. :)

Answer 12

Um, I'm not in Calculus...just in Algebra II

Answer 13

Ah, ha. I see. Ok. Well then things get a bit messier.

Answer 14

Did they WANT you to find an exact answer? Or just an approx one?

Answer 15

Exact. I'm not allowed to use a calculator here.

Answer 16

@KingGeorge It's got your name on it! Literally :)

Answer 17

I think I have a solution which involves algebra... gimmie a second to verify. :)

Answer 18

Bam. Yes. I have it. :)

Answer 19

How do you get it?

Answer 20

I'll attach a pic, hold on...

Answer 21

I'm sorry, but I have to go. Can you leave how to get it here and I'll come back to it later? Sorry :/

Answer 22

Reflect (7,8) about the x axis. The path from (4,3) to the reflection of (7,8) is the one you want.

Answer 23

You should be able to easily find the equation of the line joining (4,3) and the reflection of (7,8) about the x-axis and then find its x-intercept.

Answer 24

Oh, that makes sense! Since reflecting it over a line that you need to cross anyway doesn't affect the shortest distance, I can just reflect it so that it forms one line that performs both tasks while maintaining the same distance! Thanks! :D