Any hints would be appreciated

[Darya]

There are 2 petrol stations 6 km away from each other. At the first one, 1 liter of petrol costs 21$, at the second costs 20$. The motorist parks at the first station but decides to go to the second station, refuel and go back. At least how many liters does he have to refuel to make the journey worth it?

I don't quite understand the task. All I've come up with is an equation [MATH]-12*20+20x>0[/MATH] Then x=13, doesn't seem right tho. Any ideas?

 

[MarkFL]

Well, we know the motorist will save $1 per liter purchased, but we don't know how much fuel will be consumed to drive the 12 km to the other station and back. Suppose the motorist's automobile gets \(E\) kilometers per liter (fuel economy). Then:

[MATH]\frac{12}{E}[/MATH]
liters of fuel will be consumed making the round trip. If \(x\) is the number of liters of fuel purchased, then I would say we want the cost per liter gained to be minimized:

[MATH]\frac{20x}{x-\dfrac{12}{E}}<21[/MATH]

 

[Steven G]

Let x be the number of liters purchased.
So she saves $1, x times for a total of $x by going to the cheaper station. But the motorist had to travel an additional 12km. So we want the cost of driving those 12km to be less than the savings of $x. For example if it cost the motorist $5 to drive the 12 km, then it will not be worth the trip if she saves only $3.

So what is the cost of driving the 12km. That really depends on the kml (kilometer per liter). Suppose we call this value E. Then the amount of liters used to travel the additional 12km will be 12/E. So the cost of traveling the 12km extra will cost an extra $20*12/E

That is, we want 20*12/E<x

 

[Steven G]

Well, we know the motorist will save $1 per liter purchased, but we don't know how much fuel will be consumed to drive the 12 km to the other station and back. Suppose the motorist's automobile gets \(E\) kilometers per liter (fuel economy). Then:

[MATH]\frac{12}{E}[/MATH]
liters of fuel will be consumed making the round trip. If \(x\) is the number of liters of fuel purchased, then I would say we want the cost per liter gained to be minimized:

[MATH]\frac{20x}{x-\dfrac{12}{E}}<21[/MATH]

Mark,
At least one of us is wrong as we got different. Obviously I think that I am correct-otherwise I would not have made by post. I also had the benefit of being able to compare our answers, which you did not. I think that your 21 should be 20. Please confirm who is correct. If I am wrong then please tell me where my error is. Thanks!

 

[MarkFL]

Mark,
At least one of us is wrong as we got different. Obviously I think that I am correct-otherwise I would not have made by post. I also had the benefit of being able to compare our answers, which you did not. I think that your 21 should be 20. Please confirm who is correct. If I am wrong then please tell me where my error is. Thanks!

Okay, my inequality does reduce to:

[MATH]\frac{12\cdot21}{E}<x[/MATH]
When you give the additional cost, it appears to be in terms of the price/liter of the proximal station, rather than the distal one.

 

[Darya]

I

Well, we know the motorist will save $1 per liter purchased, but we don't know how much fuel will be consumed to drive the 12 km to the other station and back. Suppose the motorist's automobile gets \(E\) kilometers per liter (fuel economy). Then:

[MATH]\frac{12}{E}[/MATH]
liters of fuel will be consumed making the round trip. If \(x\) is the number of liters of fuel purchased, then I would say we want the cost per liter gained to be minimized:

[MATH]\frac{20x}{x-\dfrac{12}{E}}<21[/MATH]

I couldn't be thankful enough!

 

[Steven G]

Okay, my inequality does reduce to:

[MATH]\frac{12\cdot21}{E}<x[/MATH]
When you give the additional cost, it appears to be in terms of the price/liter of the proximal station, rather than the distal one.

I knew what your inequality reduced to. I still think I am correct but will think about why it should be 21 instead as 20. Thanks.

 

[MarkFL]

I knew what your inequality reduced to. I still think I am correct but will think about why it should be 21 instead as 20. Thanks.

I was wrong when I said you were basing the additional cost on the wrong station. You had that right.

What I did was to compare the cost/net liters gained in both situations. You are comparing the total cost, rather than cost/liter.