maximum minimum problems...

[Sanguinne]

the cost of running a train at a constant speed of v km/h is C = 50 + (v²/1000) dollars per hour.

a) find the time taken for an 800km journey in terms of v.
b) hence, find an expression for the cost of an 800kmn journey.
c) find the most economical speed for this jouney

i did a and b, but i cant do c. i tried to differentiate the expression for the cost and then let it equal 0 and solve v, but my answer is wrong..

[drcrowthorne]

a) So v is a speed (in kilometers an hour). Hence, to figure out the time taken to traverse a particular distance at a speed v, we have to use the following formula:

t = 800/v

b) Now, what we have to do, is to transpose the above equation, to make t the subject. This changes in the following manner:

t*v = 800/v *v

and then

(1/t)*t*v = 800*(1/t)

and becomes:

v = 800/t

We sub this into the cost equation, in place of v:

C = 50 + ((800/t)²/1000)

And that's the equation that you want. You can sort of simplify it to this:

C = 50 + (640/t²)

c) Basically all you have to find now is the lowest cost over the domain [0, infinity). Using this cost, find the amount of time it takes (say for instance it's $800 at 3 hours), and so all you do, is divide 800 by the time (3 hours) to get the optimum speed.

These questions are all about transposition of the equation, and subbing in those trasnposed equations you got. So that's all you really have to do. Hope it helps