Exponetial Decay Question?

bit of a bogey

This question may be easy for many of you but I cant seem to get my head around what way to approach it. Really starting to wreck my head!

Can anyone just rid me of this pain?

I have a sugar jar of 100gramms and today I use 4gramms of sugar in my tea. If I decrease my sugar use by 10% per day, how long will it be until I only have 1 gramm of sugar left in my jar?

Fremen

You start out with 4g on the first day. You decrease by 10% on the second day, so you use

4*(9/10)g.

On the third day, you use

4*(9/10)^2 grammes,

and so on.

You probably know the formula for summing a geometric series, so the problem reduces to finding how many terms in the above series you need to sum over to get 99.

hivizman

bit of a bogey wrote: »

I have a sugar jar of 100gramms and today I use 4gramms of sugar in my tea. If I decrease my sugar use by 10% per day, how long will it be until I only have 1 gramm of sugar left in my jar?

An interesting trick question.

Summing the geometric progression, as Fremen correctly suggests, as the number of days tends to infinity, the sum tends to 4/(1 - 0.9) = 4/0.1 = 40. However long you continue, you will never consume more than a total of 40 grams of sugar, so the quantity of sugar in the jar will never fall below 60 grams.

If you'd started with a daily consumption of 10 grams, your total consumption would tend to 10/(1 - 0.9) = 100, so it would take you infinite time to empty the jar. You would reduce the quantity to 1 gram in 44 days.

On the other hand, if you start at 4 grams per day and reduce daily consumption by 1% per day rather than 10%, you would be left with 1.89 grams of sugar after 28 days, but this would not be enough to satisfy your demand for 3.02 grams on the 29th day. These numbers come from an Excel spreadsheet rather than from solving an equation.

Fremen

Huh. I was thinking about the strategy rather than the actual numbers, so I totally missed that.

bit of a bogey

Thanks folks! Helped to clear my head on that one!