Answer :
To solve this problem, we need to analyze the given information and determine which statements correctly reflect the scenario where Marlene rides her bike at 16 miles per hour, with time [tex]$t$[/tex] representing hours and distance [tex]$d$[/tex] representing miles.
1. Statement: "The independent variable, the input, is the variable [tex]$d$[/tex], representing distance."
- In this scenario, the distance traveled depends on the amount of time Marlene rides. The time [tex]$t$[/tex] is what determines the distance [tex]$d$[/tex]. Therefore, time [tex]$t$[/tex] is the independent variable, not distance [tex]$d$[/tex]. This statement is false.
2. Statement: "The distance traveled depends on the amount of time Marlene rides her bike."
- This statement is accurate. Given that Marlene rides at a constant speed of 16 miles per hour, the distance [tex]$d$[/tex] she travels is directly dependent on the time [tex]$t$[/tex] she rides. If [tex]$t$[/tex] increases, [tex]$d$[/tex] will increase proportionately. This statement is true.
3. Statement: "The initial value of the scenario is 16 miles per hour."
- The initial value usually refers to the starting value of the distance traveled when [tex]$t=0$[/tex]. At [tex]$t=0$[/tex], the distance [tex]$d$[/tex] would be 0 miles. Therefore, this statement is referring to the speed, not the initial value of the distance traveled. This statement is false.
4. Statement: "The equation [tex]$t=d+16$[/tex] represents the scenario."
- To check the validity of this equation, we rewrite it in terms of [tex]$d$[/tex]. The relationship between distance and time should be [tex]$d = 16t$[/tex]. Solving [tex]$t = d + 16$[/tex] for [tex]$d$[/tex] does not match this relationship. Hence, this statement is mathematically incorrect and is false.
5. Statement: "The function [tex]$f(t)=16t$[/tex] represents the scenario."
- This function correctly represents the relationship between distance [tex]$d$[/tex] and time [tex]$t$[/tex]. If Marlene rides for [tex]$t$[/tex] hours at 16 miles per hour, the distance traveled [tex]$d$[/tex] is indeed [tex]$16t$[/tex]. Therefore, this statement is true.
Given this analysis, the two statements that are true are:
(2) The distance traveled depends on the amount of time Marlene rides her bike.
(5) The function [tex]$f(t)=16t$[/tex] represents the scenario.
1. Statement: "The independent variable, the input, is the variable [tex]$d$[/tex], representing distance."
- In this scenario, the distance traveled depends on the amount of time Marlene rides. The time [tex]$t$[/tex] is what determines the distance [tex]$d$[/tex]. Therefore, time [tex]$t$[/tex] is the independent variable, not distance [tex]$d$[/tex]. This statement is false.
2. Statement: "The distance traveled depends on the amount of time Marlene rides her bike."
- This statement is accurate. Given that Marlene rides at a constant speed of 16 miles per hour, the distance [tex]$d$[/tex] she travels is directly dependent on the time [tex]$t$[/tex] she rides. If [tex]$t$[/tex] increases, [tex]$d$[/tex] will increase proportionately. This statement is true.
3. Statement: "The initial value of the scenario is 16 miles per hour."
- The initial value usually refers to the starting value of the distance traveled when [tex]$t=0$[/tex]. At [tex]$t=0$[/tex], the distance [tex]$d$[/tex] would be 0 miles. Therefore, this statement is referring to the speed, not the initial value of the distance traveled. This statement is false.
4. Statement: "The equation [tex]$t=d+16$[/tex] represents the scenario."
- To check the validity of this equation, we rewrite it in terms of [tex]$d$[/tex]. The relationship between distance and time should be [tex]$d = 16t$[/tex]. Solving [tex]$t = d + 16$[/tex] for [tex]$d$[/tex] does not match this relationship. Hence, this statement is mathematically incorrect and is false.
5. Statement: "The function [tex]$f(t)=16t$[/tex] represents the scenario."
- This function correctly represents the relationship between distance [tex]$d$[/tex] and time [tex]$t$[/tex]. If Marlene rides for [tex]$t$[/tex] hours at 16 miles per hour, the distance traveled [tex]$d$[/tex] is indeed [tex]$16t$[/tex]. Therefore, this statement is true.
Given this analysis, the two statements that are true are:
(2) The distance traveled depends on the amount of time Marlene rides her bike.
(5) The function [tex]$f(t)=16t$[/tex] represents the scenario.
