Question

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.

Answer 1

Answer:

0 ≤ t ≤ 5.

Step-by-step explanation:

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

In this case, $f(t)$ 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,

• $t \ge 0$.
• $f(t) \ge 0$.

The first condition is an inequality about $t$, which is indeed the independent variable.

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

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

Hence, t ≤ 5.

Combine the two inequalities to obtain the domain:

0 ≤ t ≤ 5.