1 Attachment(s)

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.

Attachment 23934

Re: Find shortest time to travel

Hello, ineedhelplz!

The diagram and description are not correct.

I think I've corrected them,

Quote:

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.

. . $\displaystyle AC = 2,\;BC = d$

He walks through the "forest" from point A to point P

. . then walks in the "clear" from point P to point B.

Let $\displaystyle x = CP;\;d-x = PB.$

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

. . This takes $\displaystyle \frac{\sqrt{x^2+4}}{3}$ hours.

He walks distance $\displaystyle PB = d-x$ at 5 km/hr.

., . This takes $\displaystyle \frac{d-x}{5}$ hours.

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

And *that* is the function we must minimize.,

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