Pythagorean theorem: minimizing time, using boat, walking

[edgarrapadas]

Hi,
Below is an assignment of my son. can you provide me the step by step solution?

A boatman is at A which is 4.5km from the nearest point B on a straight shore BM. He wishes to reach in minimum time at point C situated on the shore 9 km from B. How far from C should he land if he can row at the rate of 6km/hr and can walk at the rate of 7.5 km.hr? Ans. 3 km

 

[tkhunny]

Re: Pythagorean theorem

Are you home schooling? Rule #1 for home schooling - If you can't handle the math, you WILL have a problem. Get local outside help sooner than later.


AB = 4.5 km

BC = 9 km

Put point D between B and C.

The question asks, what distance DC will minimize the travel time.

The Pythagorean Problem is \(\displaystyle AD^{2} = AB^{2} + BD^{2} ==> AD = \sqrt{AB^{2} + BD^{2}}\)

Now the last piece: Distance = Rate * Time or Time = Distance/Rate

This makes total travel time: AD/(6 km/h) + DC/(7.5 km/h)

One more thing. We know that BD + DC = 9 km -- Time for multiple substitutions...

\(\displaystyle \sqrt{AB^{2} + BD^{2}}/(6 km/h) + DC/(7.5 km/h)\)

\(\displaystyle \sqrt{(4.5 km)^{2} + BD^{2}}/(6 km/h) + DC/(7.5 km/h)\)

\(\displaystyle \sqrt{(4.5 km)^{2} + BD^{2}}/(6 km/h) + (9 km - BD)/(7.5 km/h)\)

And there you have it. Everything is in terms of BD and you should be able to find its minimum value.

 

[stapel]

Below is an assignment of my son. can you provide me the step by step solution?

Um... Shouldn't your son be doing his assignment? I mean, okay, you're not in his class, so you don't know the material. But us doing the assignment for you, so you can do it for him, isn't likely to do anything productive. Sure, he'll get full points on this assignment (if somebody here completes it to his teacher's satifaction, and you copy it out accurately), but--

It might be better if your son posted his work and reasoning, showing what he'd tried and where he is stuck. Then we could help him actually learn. If that is of interest to you.

Thank you!

Eliz.