Hard Q, Help Now

[mmulv2]

I need help please. Thankyou.

A bushwalker can walk at 5km/h through clear land and 3km/h through bushland. If she has to het from point a to point b following a route indicated in the image, find the value of x so that the route is covered in a minimum time. (Note: time = distance/speed)

Image: http://img294.imageshack.us/my.php?i...ntitledtp0.jpg

Please help, i need helkp getting to the answer. Please post all working. thnks

[krtxmrtz]

Welcome to the forums.

I'll give you the general solution and then will substitute the actual values. Let A and B the horizontal and vertical distances respectively from a to b. Call v₁ and v₂ the walking velocities in the bush and clear respectively and t₁ and t₂ the times taken to cross each region.

The total time to be minimized is
T = t₁ + t₂ = x₁ / v₁ + x₁ / v₂

Now, x₁ = Sqr(a² + x²) and x₂ = b - x. Then,

T = Sqr(a² + x²) / v₁ + (b - x) / v₂

For the x value that minimizes T, the derivative of T with respect to x must be 0:

0 = dT/dx = -1/v₂ + x/v₁Sqr(a² + x²)

From here and with some algebra,

x = a/Sqr([v₂/v₁]² - 1)

Finally,

v₁ = 3 km/h
v₂ = 5 km/h
a = 2 km
b = 3 km

This yields: x = 1.5 km