Question

question 4
scenario: a pot of soup is taken off a hot stove and allowed to cool. consider the temperature as a function of time since the soup was taken off the stove.
determine a reasonable domain and explain it in complete sentences.
determine a reasonable range and explain it in complete sentences.

Explanation:

Step1: Define domain concept

Domain is the set of all possible input values. Here, the input is time since the soup was taken off the stove.

Step2: Determine domain values

Time cannot be negative in this real - world context. The soup starts cooling at time $t = 0$ and continues cooling indefinitely. So, a reasonable domain is $t\geq0$, where $t$ is in minutes (or any appropriate time unit).

Step3: Define range concept

Range is the set of all possible output values. The output is the temperature of the soup.

Step4: Determine range values

The initial temperature of the soup when taken off the stove is some high value (close to the boiling point of water if the soup is mostly water - based). As it cools, it will approach the ambient temperature of the room. So, a reasonable range is $T_{ambient}\leq T\leq T_{initial}$, where $T$ is the temperature of the soup, $T_{ambient}$ is the ambient temperature of the room, and $T_{initial}$ is the initial temperature of the soup when taken off the stove.

Answer:

Domain: $t\geq0$ (in appropriate time units). Explanation: Time since the soup was taken off the stove cannot be negative; it starts at 0 and goes on.
Range: $T_{ambient}\leq T\leq T_{initial}$. Explanation: The temperature of the soup starts at its initial high value when taken off the stove and will cool down to approach the ambient temperature of the room.