Math Is Fun Forum

uzurpatorul

Member

Registered: 2009-06-25

Posts: 11

Crossing the dessert

An  unlimited  supply  of  gasoline  is  available  at  one  edge  of  a  desert  800  miles  wide,  but  there  is  no  source  in  the  desert  itself.  A  truck  can  carry  enough  gasoline  to  go  500  miles  (this  is  called  the  "load"),  and  it  can  build  its  own  refueling  stations  at  any  spot  along  the  way.

What  is  the  minimum  amount  of  gasoline (in loads)  the  truck  will  require  in  order  to  cross  the  desert?

Last edited by uzurpatorul (2009-06-27 06:23:59)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Crossing the dessert

with n = # of loads of fuel

when n = 3   then 766.666 miles are traveled.
when n = 4   then 838.095 miles are traveled.
So the amount of fuel required is between 3 and 4 loads.

We need an analytical form for the series but it is complicated and involves the psi function. We make do with an approximation to the sum.

A)

Solving for n:

n = 3.4435 loads of fuel necessary. When I bring this answer to the site, it still doesn't work. Possibly the approximation of the sum used in A is causing the error.  When I sum the series exactly I get n = 3.4315, pretty close. This is out of Numerical Analysis by Francis Scheid.

Last edited by bobbym (2009-06-25 22:13:20)

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

quittyqat

Member

Registered: 2009-04-08

Posts: 1,215

Re: Crossing the dessert

Someone who has the ability may want to change the title from Crossing the dessert to Crossing the desert.

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I'll be here at least once every decade.

uzurpatorul

Member

Registered: 2009-06-25

Posts: 11

Re: Crossing the dessert

You general solution is right, however final answer is little over

.

P.S.

I guess I was craving for sweets when I spelled the title.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Crossing the dessert

Hi uzurpatorul;

Can you show me your calculations because since my method is correct I think that "hidden text" answer of 3.46 is not correct. I am very sure of  ≈ 3.4315 .

Last edited by bobbym (2009-06-26 05:41:26)

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

uzurpatorul

Member

Registered: 2009-06-25

Posts: 11

Re: Crossing the dessert

Lets assume that your solution is the right one, and indeed with 3.44 loads we can cross 800 miles:

with 1 load , we can go 500 miles
with 2 loads , we can go 500(1+1/3) = 666.66
with 3 loads, 500(1+1/3+1/5) = 766.66

now, it means that 3.44 - 3 = 0.44 loads we can get (800 - 766.66)x7 (we need 7 trips now to make a cache of 3 loads)

0.44 x 500miles/load = 220miles , (800 - 766.66)x7 = 233.38

0.4666 x 500miles/load = 233.3 (much closer to the answer)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Crossing the dessert

Hi uzurpatorul;

Hard to argue with that. Nice and neat proof. So if my sum is okay then the problem must be with solving it analytically and setting it equal to 800.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

uzurpatorul

Member

Registered: 2009-06-25

Posts: 11

Re: Crossing the dessert

if you look at my explanation is pretty much your formula for n between 3 and 4

500(1 + 1/3 + 1/5 + k/7) = 800, where k = n - 3

Another cool thing i noticed looking at your formula is that you just proof that the series is divergent, and with the common constrains any desert can be crossed (of course n grows exponentially).

P.S.

here are 2 variations of the same problem:
http://www.projecteureka.org/problem/question/105
http://www.projecteureka.org/problem/question/298

Last edited by uzurpatorul (2009-06-27 06:23:24)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Crossing the dessert

Hi uzurpatorul;

Thanks for links and the education.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.