Optimization Q ?

SmokyMo

I am really stuck on this one. Help needed......

A company salesman is to travel from A to B at a constant speed of x km/hour. The journey is to be taken using a car which will travel a distance of 400x/(x2+100) km per litre of petrol (when its speed is x km per hour).If petrol costs 1 euro per litre and the salesperson is paid a flat wage of 20 euro per hour, do the following

1.Determine the petrol costs per km.

2.Determine the wage costs per km.

3.Determine the constant speed,x,at which the salesman should drive in order to minimize the total cost of the journey to the company.

Thanks

Fremen

Well, the slower the car goes, the cheaper it is to get from A to B. However, if the car goes slower, the company needs to pay the driver more because he'll be travelling for longer.

If D is the distance between A and B, and if the salesman does the trip in one hour, how much will it cost (in terms of D, since you are not given D)? what if he does it in two hours? See if you can generalise to N hours.

SmokyMo

My brain is melting away looking at this q for over an hour.........

cavedave

This might help
http://www-128.ibm.com/developerworks/linux/library/l-glpk1/

Fremen

Let D be the distance from A to B

If the salesman travels at V kph:

He will complete the journey in D/V hours, which will cost 20D/V euro in wages

We know that he can travel a distance 400V/(2V + 100) on one euro's worth of petrol
Thus, he can travel one mile on (2V + 100)/400V euro's worth of petrol
So he can travel D miles on D(2V + 100)/400V euro's worth of petrol

Then the total cost of the trip is

D(2V + 100)/400V + (20D/V)

so find the value of V which minimises that expression, and everything else should follow.

Edit: feck, that's a strictly decreasing function of V, so I've gone wrong somewhere

MathsManiac

I don't think you need to introduce V there; you can do it all with x.

You're given the km/litre, so just turn it upside down to get the litres/km (which is the same as fuel cost per km, since 1 litre costs €1 [ah, those were the days...])

Wage costs €20 per hour, so it'll be €20/x per km.

The total cost per km is the sum of these two.

Then minimise it. You should get 90 km/hr, I think.

SmokyMo

The total cost per km (x^2-8100)/x2 How do I minimise it in order to get 90?

Grudaire

SmokyMo wrote: »

The total cost per km (x^2-8100)/x2 How do I minimise it in order to get 90?

Differentiate?

SmokyMo

Cliste wrote: »

Differentiate?

It doesnt make any sense to differentiate agan because i will get 2x/2x

MathsManiac

SmokyMo wrote: »

The total cost per km (x^2-8100)/x2 How do I minimise it in order to get 90?

Sorry for delay - I've been away.

The total cost per km is (x^2+100)/(400x) + 20/x,

which equals (x^2+8100)/(400x).

You can diff that using the quotient rule, or rewrite it as: x/400 + (81/4)x^-1.

Diff to get: 1/400 - (81/4)x^-2

Set that equal to zero; then multiply across by 400x^2 to get x^2-8100=0, giving x=90km/h.

(And diff again to verify it's a min., if you're expected to do that.))