Thread: Find shortest time to travel

1. Find shortest time to travel

Firstly, apologies for the hard-to-see image. It's just for an outline of the problem.

The questions asks to find the value of x (on the right vertical side) to find the minimum time to travel to the top right corner of the "forest"

The bush can be travelled through at 3km/h and the "clear" area can be travelled through at 5km/h.

I started by equating the times of the 2 routes to take but came up with the wrong answer.

Any help is greatly appreciated.

The graphic shows 2 km at the bottom and x km on the right side.

2. Re: Find shortest time to travel

Hello, ineedhelplz!

The diagram and description are not correct.
I think I've corrected them,

Find the value of x (on the right vertical side) to find the minimum time to travel to the top right corner of the "forest".
The forest can be travelled through at 3 km/h and the "clear" area can be travelled through at 5 km/h.

The graphic shows 2 km at the bottom and x km on the right side.

Code:
* B -
|   :
|d-x:
|   :
*P  :
*  |   d
*      |   :
*        |x  :
*           |   :
*              |   :
A * - - - - - - - - *C  -
: - - -  2  - - - :
A man wants to get from A to B in the least amount of time.
. . $AC = 2,\;BC = d$

He walks through the "forest" from point A to point P
. . then walks in the "clear" from point P to point B.
Let $x = CP;\;d-x = PB.$

He walks distance $AP = \sqrt{x^2+4}$ at 3 km/hr.
. . This takes $\frac{\sqrt{x^2+4}}{3}$ hours.

He walks distance $PB = d-x$ at 5 km/hr.
., . This takes $\frac{d-x}{5}$ hours.

The total time is: . $T \:=\:\tfrac{1}{3}(x^2+4)^{\frac{1}{2}} + \tfrac{d}{5} - \tfrac{1}{5}x$

And that is the function we must minimize.,

3. Re: Find shortest time to travel

Thank you very much, that makes a lot of sense, however; how do I factor the extra distance d into the equation?

I get an answer of x=1.5 which is what I got initially. The answer in the solutions is 1.2km.

Could you please further complete this example? That would be a great help!

Thank you