Applications of an Exponential Function

Example 1
After starting an experiment, the number of bacteria in a culture is given by the expression:
 N(t) = 1200*2^0.4t
How long after the start of the experiment will the culture have 19200 bacteria?
N(t) = 1200*2^0.4t
N(t) = 19200
1200*2^0.4t = 19200
2^0.4t = 19200/1200
2^0.4t = 16
2^0.4t = 2⁴
0.4t = 4
t = 4/0.4
t = 10 h
The culture will have 19200 bacteria after 10 h.
Example 2
The amount of R$ 1200.00 was applied for 6 years in a banking institution at a rate of 1.5% per month, in the compound interest system.
a) What will the balance be at the end of 12 months?
b) What will the final amount be?
M = C(1+i)^t (Compound interest formula) where:
C = capital
M = final amount
i = unit rate
t = application time
a) After 12 months.
Resolution
M = ?
C = 1200
i = 1.5% = 0.015 (unit rate)
t = 12 months
M = 1200(1+0.015)¹²
M = 1200(1.015) ¹²
M = 1200*(1.195618)
M = 1,434.74
After 12 months he will have a balance of R$1,434.74.
b) Final amount
Resolution
M = ?
C = 1200
i = 1.5% = 0.015 (unit rate)
t = 6 years = 72 months
M = 1200(1+ 0.015)

⁷²
M = 1200(1.015) ⁷²
M = 1200(2.921158)
M = 3,505.39
After 6 years he will have a balance of R$ 3,505.39
Example 3
Under certain conditions, the number of B bacteria in a culture, as a function of time t, measured in hours, is given by B(t) = 2^t/12. What will be the number of bacteria 6 days after the zero hour?
6 days = 6 * 24 = 144 hours
B(t) = 2^t/12
B(144) = 2^144/12
B(144) = 212
B(144) = 4096 bacteria
The culture will have 4096 bacteria.

by Mark Noah
Graduated in Mathematics
Brazil School Team

Source: Brazil School - https://brasilescola.uol.com.br/matematica/aplicacoes-uma-funcao-exponencial.htm