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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][tex]$d$[/tex][/tex]. This relationship is modeled with distance [tex][tex]$d$[/tex][/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][tex]$d$[/tex][/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][tex]$t=d+16$[/tex][/tex] represents the scenario.

E. The function [tex][tex]$f(t)=16t$[/tex][/tex] represents the scenario.



Answer :

GinnyAnswer
To solve this question, we need to analyze the relationship between the distance [tex]\( d \)[/tex] and the time [tex]\( t \)[/tex] that Marlene rides her bike.

1. Statement Analysis:

- The independent variable, the input, is the variable [tex]\( d \)[/tex], representing distance.
- This statement is incorrect. In this scenario, the time [tex]\( t \)[/tex] Marlene rides her bike is the independent variable (input), and the distance [tex]\( d \)[/tex] traveled is the dependent variable (output).

- The distance traveled depends on the amount of time Marlene rides her bike.
- This statement is correct. The distance [tex]\( d \)[/tex] is a function of time [tex]\( t \)[/tex]. As Marlene rides her bike for more time, the distance she travels increases.

- The initial value of the scenario is 16 miles per hour.
- This statement is incorrect. The rate of 16 miles per hour is not an initial value; instead, it is the rate at which Marlene rides her bike. An initial value would refer to a starting distance at time [tex]\( t = 0 \)[/tex], which in this context is zero.

- The equation [tex]\( t = d + 16 \)[/tex] represents the scenario.
- This statement is incorrect. The correct mathematical relationship should be distance [tex]\( d \)[/tex] equals the rate (16 miles per hour) times the time [tex]\( t \)[/tex]. The equation provided does not correctly represent this relationship.

- The function [tex]\( f(t) = 16t \)[/tex] represents the scenario.
- This statement is correct. The distance [tex]\( d \)[/tex] can be represented by the function [tex]\( f(t) = 16t \)[/tex], indicating that the distance traveled by Marlene is the product of her rate (16 miles per hour) and the time [tex]\( t \)[/tex] in hours.

2. Conclusions:

Based on the analysis, the two statements that are true for the scenario are:

1. The distance traveled depends on the amount of time Marlene rides her bike.
2. The function [tex]\( f(t) = 16t \)[/tex] represents the scenario.

These two statements accurately describe the relationship between the time Marlene rides her bike and the distance she covers.

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