Asubstance decays so that the amount a of the substance left after t years is given by: a = a 0 · (0.75) t , where a 0 is the original amount of the substance. what is the amount of time it takes for there to be one-third of this substance left, rounded to the nearest tenth of a year?

Let  be the original amount of the substance. After t years, the amount left A is given by:

                               .

For example, after 1 year, the amount is

      .

Thus, after one year, only 0.75=3/4 of the original amount is left.

We want to solve for t, such that   is one-third of , so we set the equation:

.

Simplifying by , we have:

.

This is an exponential equation, so we can solving it by rewriting it as a logarithm:

.

3.8

Each friend will get 1.7 of a pack of cards