One day at school, 3 kids are infected with a flu virus. The next day, only two of them return to school. At the end of each day, the number of infected students increases by 40%. Write a formula that represents the total number of infected students in the form of $f \left(t\right) = a {b}^{t} + c$.

Apr 19, 2017

Let $f \left(t\right)$ be the total number of infected students at time $t$.
Then:

$f \left(t\right) = 2 {\left(1.4\right)}^{t} + 1$

Explanation:

The function value $a$ represents the seed value; it is the number from which the exponential growth will start. For this question, that is the 2 students who return to school the next day. They will be the ones who spread the flu virus.

$a = 2$

The function value $b$ represents the rate of growth for every unit of time. For this question, that value is given as the 40% increase. This is written as 1.4, because the number of infected students increases by 40% each day, which is done by multiplying the current number of infected students by 140% (or 1.4).

$b = 1.4$

The function value $c$ represents the initial constant, a quantity that does not contribute to the growth, but does contribute to the overall total. For this question, that would be the one student who stayed home. He counts towards the total number of sick students, but since he is not in school, he does not contribute to the infection of the other students.

$c = 1$

Note:

$a$ has the units of "students"
$b$ is a scalar (it has no units)
$c$ has the units of "students"

Thus, the units of the function $f \left(t\right)$ work out as:

$\left[\left(f \left(t\right)\right) , \left(\text{students")]=[(2),("students")][(1.4),(@)]^t+[(1),("student}\right)\right]$

or

$\text{students = students + students}$

So the units make sense.