Exponential models

If something changes, by a certain percent at the same time, the change is exponential. It can be an increase, such as an increase in animal population or the balance of a savings account that pays interest annually, or it can be a decrease, such as radioactive decay or the amount of medicine in the body.

Example 1

A deposit of €1000 is made to a savings account. The deposit pays an annual interest rate of 2,0%.

An exponential model is constructed to describe the amount of the deposit after x years.

The deposit becomes 1,02 times greater each year. The money in the account is multiplied by 1,02 annually.

We form a function f, which expresses the amount of money in the account after x years.

Now, for example, we can find out how much money is in the account after 10 years

Or how many years does it take for the account to have 2000 €?

Therefore, it would take 35 years for the account balance to reach €2000

Exponential growth often becomes very large, very quickly.

Example 2

Liisa-Petter got a job as an agricultural farmhand even though she had almost no experience. The matter of pay was decided by a wage request. Liisa-Petter asked for a salary of €1 for the first week and €2 for the following week and then for the salary to be doubled every week for three months. After this period, it would remain the same. Ah, that's nothing, thought the employer and quickly signed the contract.

How much did Liisa-Petter's weekly salary eventually become after three months?

For the first week, the salary is €1 and this will double 12 times, so the salary will be

The starting salary of 1 euro turned into a salary of €4096 in just three months. This was a weekly salary, so Lisa-Peter received €16 384 a month.

Example 3

The amount of caffeine absorbed into the blood is halved about every 5 hours. One 1,5 litre bottle of cola drink contains 195 milligrams of caffeine. Caffeine is rapidly absorbed into the blood, and the maximum effect is seen in about half an hour. It is assumed that after half an hour all 195 mg of caffeine is absorbed into the blood.

How much caffeine is in the blood 10 hours after the maximum effect?

In ten hours, the level is halved twice, leaving a quarter of the caffeine, or 48,75 mg.

A model describing the halving of caffeine would be

In the function t is the time from the maximum effect. So, after 10 hours

With the function, we can find out, for example, how long it takes for less than 10 mg of caffeine to remain.

For less than 10 mg of caffeine remaining, it takes about 21 and a half hours.

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