Let's analyze each statement one by one:
1. The independent variable, the input, is the variable [tex]\( d \)[/tex], representing distance.
- In this scenario, the time [tex]\( t \)[/tex] is the independent variable because it is what Marlene can control or choose. The distance traveled [tex]\( d \)[/tex] depends on the time spent riding. Hence, this statement is false.
2. The distance traveled depends on the amount of time Marlene rides her bike.
- This statement accurately describes the relationship because the distance [tex]\( d \)[/tex] is a function of time [tex]\( t \)[/tex]. The longer Marlene rides, the farther she will travel. Therefore, this statement is true.
3. The initial value of the scenario is 16 miles per hour.
- The phrase "initial value" typically refers to a starting point or value at time [tex]\( t = 0 \)[/tex]. However, 16 miles per hour is the rate at which Marlene rides, not an initial value. Therefore, this statement is false.
4. The equation [tex]\( t = d + 16 \)[/tex] represents the scenario.
- This equation implies an incorrect relationship between time [tex]\( t \)[/tex] and distance [tex]\( d \)[/tex]. The correct relationship is that distance [tex]\( d \)[/tex] is equal to the rate (16 miles per hour) multiplied by the time [tex]\( t \)[/tex]. Hence, this statement is false.
5. The function [tex]\( f(t) = 16t \)[/tex] represents the scenario.
- This function correctly models the scenario where [tex]\( f(t) \)[/tex] represents the distance [tex]\( d \)[/tex] traveled, which is the product of the rate (16 miles per hour) and the time [tex]\( t \)[/tex]. Therefore, this statement is true.
Based on the analysis, the two statements that are true for the scenario are:
- The distance traveled depends on the amount of time Marlene rides her bike.
- The function [tex]\( f(t) = 16t \)[/tex] represents the scenario.
So, the correct answers are:
- The distance traveled depends on the amount of time Marlene rides her bike.
- The function [tex]\( f(t) = 16t \)[/tex] represents the scenario.
