Is the number of gallons of water in the bathtub a function of time?
A. Yes because each [tex]$t$[/tex] value is paired with exactly one number of gallons.
B. Yes because the number of gallons is bounded between [tex]$y=0$[/tex] and [tex]$y=36$[/tex].
C. No because each number of gallons is paired with more than one [tex]$t$[/tex] value.
D. No because from [tex]$t=5$[/tex] to [tex]$t=20$[/tex] the number of gallons of water is constant.
Answer (2)
A function requires each input (time) to have only one output (number of gallons).
Option 1 states that each time value is paired with exactly one number of gallons, which aligns with the definition of a function.
The other options provide irrelevant or contradictory justifications.
Therefore, the correct answer is: Yes because each t value is paired with exactly one number of gallons.
Explanation
Understanding the Problem The question asks whether the number of gallons of water in a bathtub is a function of time. To determine this, we need to understand the definition of a function. A function requires that each input (in this case, time) has only one output (the number of gallons of water). We need to analyze the given justifications to see which one correctly applies this definition.
Analyzing the Options Let's analyze each option:
Option 1: Yes because each t value is paired with exactly one number of gallons. This statement aligns with the definition of a function. If each moment in time corresponds to a single, specific amount of water in the bathtub, then the number of gallons is indeed a function of time.
Option 2: Yes because the number of gallons is bounded between y = 0 and y = 36 . The fact that the number of gallons is bounded doesn't determine whether the relationship is a function. A function can have bounded or unbounded outputs. This option is irrelevant.
Option 3: No because each number of gallons is paired with more than one t value. This statement contradicts the definition of a function. If a specific amount of water in the bathtub occurs at multiple times, it doesn't necessarily mean it's not a function. The key is whether each time has only one corresponding amount of water.
Option 4: No because from t = 5 to t = 20 the number of gallons of water is constant. If the number of gallons remains constant over a period of time, it simply means that the rate of change is zero during that interval. This does not violate the definition of a function, as each time t still corresponds to only one value for the number of gallons.
Conclusion Based on the definition of a function, the correct answer is:
Yes because each t value is paired with exactly one number of gallons.
Examples
Consider tracking the temperature of a room over time. If at every specific time, there is only one temperature reading, then the temperature is a function of time. This concept is used in weather forecasting, where temperature, humidity, and wind speed are all functions of time, allowing meteorologists to predict future weather conditions. Similarly, in economics, the price of a stock can be seen as a function of time, with each moment corresponding to a unique price.
The correct answer is Option A: Yes because each t value is paired with exactly one number of gallons. This aligns with the definition of a function, where each input corresponds to a unique output. Hence, the number of gallons in the bathtub is indeed a function of time.
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