A swimming pool is being drained so it can be cleaned. The amount of water in the pool is changing according to the función f(t)= 80,000 - 16,000t , where t = time in hours and f(t) = amount of water in liters. What is the domain of this function in this situation? Explain how you found your answer.

In the function [tex]f(t)[/tex], [tex]t[/tex] is the independent variable. The domain of [tex]f[/tex] is the set of all values of [tex]t[/tex] that this function can accept.

In this case, [tex]f(t)[/tex] is defined in a real-life context. Hence, consider the real-life constraints on the two variables. Both time and volume should be non-negative. In other words,

[tex]t \ge 0[/tex].
[tex]f(t) \ge 0[/tex].

The first condition is an inequality about [tex]t[/tex], which is indeed the independent variable.

However, the second condition is about [tex]f[/tex], the dependent variable of this function. It has to be rewritten as a condition about [tex]t[/tex].

[tex]\begin{aligned} f(t) &\ge 0 &&\text{Assumption} \cr 80000 - 16000\, t& \ge 0 && \text{Definition of} ~ f \cr 80000 & \ge 16000\, t && \begin{aligned}&\text{Add $16000\, t$} \\[-0.5em] & \text{to both sides of the inequality}\end{aligned} \cr 5 &\ge t &&\begin{aligned}&\text{Divide both sides of} \\[-0.5em] & \text{the inequality by $16000$}\end{aligned} \cr t &\le 5 && \text{Flip the inequality}\end{aligned}[/tex].

Hence, t ≤ 5.

Combine the two inequalities to obtain the domain:

0 ≤ t ≤ 5.