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Marlene rides her bike at a rate of 16 miles per hour. The time in hours that she rides is represented by the variable [tex][tex]$t$[/tex][/tex], and the distance she rides is represented by the variable [tex]$d$[/tex]. This relationship is modeled with distance [tex]$d$[/tex] as a function of time [tex][tex]$t$[/tex][/tex].

Which statements are true of the scenario? Select two answers.

A. The independent variable, the input, is the variable [tex]$d$[/tex], representing distance.
B. The distance traveled depends on the amount of time Marlene rides her bike.
C. The initial value of the scenario is 16 miles per hour.
D. The equation [tex]$t=d+16$[/tex] represents the scenario.
E. The function [tex][tex]$f(t)=16t$[/tex][/tex] represents the scenario.

Sagot :

GinnyAnsweridn
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.
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