# Dishwashing mathmatical help

• Jun 6th 2010, 05:48 AM
StrugglingMathmatician
Dishwashing mathmatical help needed!
Hey, I'm new here, because I dont no where else to go with my bemusing problem...

I'm taking part in a Science Technology Engineering and Mathmatics course, but this problem which I came across is something I haven't covered within my skillset yet...

Firstly, the basic problem is about Re-usable cups vs Disposable cups, but I'm having problems doing a certian energy calculation.

I worked out that the energy cost for a re-usable cup would be:
n/x + x(r) with x being number of uses/washes, r being the energy to wash a cup in a dishwasher in joules (total dishwasher energy/total number of cups) and n being the production energy, also in joules. However, I know some of these values, making the problem:

6,300,000/x + x(64,800) = the total energy (which I will call T)

Now what I got stuck on is this...

I know it takes 310,000 joules of energy to make a plastic disposable cup. I want to find out how many times I need to use and wash the cup to get a payback time on my re-using.

So I got T=6,300,000/x + x(64,800)
But with T being 310,000

310,000 = 6,300,000/x + 64800x

But how do I find x?

I end up with:

6,300,000/x + x(64,800) = 310,000

6,300,000/x + 64,800x = 310,000

64,800x + 6,300,000/x = 310,000

64,800x = 310,000 - 6,300,000/x

and I just stop...

Have I gone wrong somewhere?
• Jun 6th 2010, 05:59 AM
SpringFan25
i have not checked your derivation, but you can solve the equation you got stuck on by multiplying both sides by x.
This gives
$\displaystyle 64,800x^2 = 310,000x - 6,300,000$
$\displaystyle 64,800x^2 - 310,000x + 6,300,000 = 0$

This is a quadratic equation that you can solve using the quadratic formula.
• Jun 6th 2010, 06:15 AM
StrugglingMathmatician
So I put that into a quadratic calculator (because I dont know how to do them) and...

Input MUST have the format:
AX2 + BX + C = 0

EXAMPLE: input the equation
2X2 + 4X -30 = 0 as:
A= 2 B= 4 C= -30

click E N T E R and your answers should be 3 and -5.

Note: Do not enter commas.

A= 64800 B= -310000 C= 6300000

X1= 2.392 plus 9.5656 i
X2= 2.392 minus 9.5656 i
Here's the answer in fractional form:
X= { 310000 ± Sq Root ( 9.6100e+10 - 1.6330e+12 ) } / 129600

From here, I'm not sure how to get to an answer of how many times I need to re-use a cup and wash it to equal 310,000 joules of energy per use.
• Jun 6th 2010, 06:26 AM
SpringFan25
Those outputs mean that the quadratic equation you derived has no solution in real numbers. In the context of this problem, it means that the equation you derived is not correct.

Lets start over from the beginning.

Energy to produce reusable cup: 6,300,000
Energy to wash reusable cup: 64,800

I'll assume you wash it before using it the first time.

Find the Total energy used once you have used the cup x times. This is the production cost plus the cost of washing it x times.
$\displaystyle 6,300,000 + 64,800x$

Now find the total energy needed to use x disposable cups
$\displaystyle 310,000x$

All you want to do is find the value of x that makes these two figures equal.
$\displaystyle 310,000x = 6,300,000 + 64,800x$
Can you solve this equation for x?

Spoiler:

Take 64800x from both sides

$\displaystyle 310,000x - 64800x = 6,300,000$
$\displaystyle 246,000x = 6,300,000$
$\displaystyle x = 25.609$

so you will be better off if you use the cup at least 26 times

• Jun 6th 2010, 06:37 AM
StrugglingMathmatician
Quote:

Originally Posted by SpringFan25
Those outputs mean that the quadratic equation you derived has no solution in real numbers. In the context of this problem, it means that the equation you derived is not correct.

Lets start over from the beginning.

Energy to produce reusable cup: 6,300,000
Energy to wash reusable cup: 64,800

I'll assume you wash it before using it the first time.

Find the Total energy used once you have used the cup x times. This is the production cost plus the cost of washing it x times.
$\displaystyle 6,300,000 + 64,800x$

Now find the total energy needed to use x disposable cups
$\displaystyle 310,000x$

All you want to do is find the value of x that makes these two figures equal.
$\displaystyle 310,000x = 6,300,000 + 64,800x$
Can you solve this equation for x?

I'm not quite sure what you mean sorry...

I'll show you what I've done so far:

After 1 use:

n = 6,300,000 and x= 1
(6,300,000/1) + 1(64800)
Total energy = 6,364,800

After 2 uses:

n = 6,300,000 and x = 2
(6,300,000/2) + 2(64800)
Total energy = 3,279,600

After 5 uses:

n = 6,300,000 and x= 5
(6,300,000/5) + 5(64800)
Total energy = 1,584,000

I want to solve the problem: If total energy = the energy it takes to produce a disposable cup (310,000), what is x (how many times must I use a re-usable cup for it to match the energy cost for 1 use of a disposable cup)

I'm pretty sure I should divide something which I haven't...
• Jun 6th 2010, 06:40 AM
SpringFan25
it is not correct to divide 6,300,000 by x and multiply 68400 by x. Depending on how you approach the question, you should do one or the other.

6,300,000 is the production cost of the cup, and it is constant. it doesn't decrease because you washed the cup 1 more time.

I dont think you have written the question correctly either
Quote:

how many times must I use a re-usable cup for it to match the energy cost for 1 use of a disposable cup)
Do you mean
how many times must I use a re-usable cup for the average energy cost to match the energy cost for 1 use of a disposable cup?

in this case, the average cost per use of the disposable cup is
$\displaystyle \frac{6,300,000}{x} + \frac{64800x}{x}$

$\displaystyle = \frac{6,300,000}{x} +64800$

not $\displaystyle \frac{6,300,000}{x} + 64800x$
• Jun 6th 2010, 06:50 AM
StrugglingMathmatician
Quote:

Originally Posted by SpringFan25
it is not correct to divide 6,300,000 by x and multiply 68400 by x. Depending on how you approach the question, you should do one or the other.

6,300,000 is the production cost of the cup, and it is constant. it doesn't decrease because you washed the cup 1 more time.

I dont think you have written the question correctly either

Do you mean
how many times must I use a re-usable cup for the average energy cost to match the energy cost for 1 use of a disposable cup?

in this case, the average cost per use of the disposable cup is
$\displaystyle \frac{6,300,000}{x} + \frac{64800x}{x}$

$\displaystyle = \frac{6,300,000}{x} +64800$

not $\displaystyle \frac{6,300,000}{x} + 64800x$

Yes, thanks, I've realised my mistake now.

[n + x(64,800)]/x with x being number of uses/washes, and n being the production energy

Cancel to 6300000/x+ 64800
• Jun 6th 2010, 07:32 AM
Mukilab
Shouldn't it be

'Struggling Mathematician'?