To determine which statements are true for the given scenario, let's analyze each statement one by one in the context of Marlene riding her bike at a steady rate of 16 miles per hour.
1. The independent variable, the input, is the variable [tex]\(d\)[/tex], representing distance.
This statement is false. In this scenario, the independent variable is time [tex]\( t \)[/tex], which serves as the input. The distance [tex]\( d \)[/tex] depends on the amount of time she rides, making distance the dependent variable. Therefore, the input in this function or scenario is time, not distance.
2. The distance traveled depends on the amount of time Marlene rides her bike.
This statement is true. The distance [tex]\( d \)[/tex] is determined by how much time [tex]\( t \)[/tex] Marlene rides the bike. The longer she rides, the further the distance she travels. Hence, distance is directly dependent on time in this context.
3. The initial value of the scenario is 16 miles per hour.
This statement is true but slightly misleading in its wording. The value of 16 miles per hour indicates the rate at which Marlene rides her bike; it is not strictly an "initial value," but it does represent the constant speed she maintains.
4. The equation [tex]\( t = d + 16 \)[/tex] represents the scenario.
This statement is false. The correct relationship between distance [tex]\( d \)[/tex] and time [tex]\( t \)[/tex] for Marlene riding her bike at 16 miles per hour would be [tex]\( d = 16t \)[/tex]. The equation [tex]\( t = d + 16 \)[/tex] does not correctly model the situation as it does not conform to the real-world relationship described.
5. The function [tex]\( f(t) = 16 t \)[/tex] represents the scenario.
This statement is true. The function [tex]\( f(t) = 16t \)[/tex], where [tex]\( t \)[/tex] is the time in hours and [tex]\( f(t) \)[/tex] is the distance, correctly represents Marlene riding her bike at a constant rate of 16 miles per hour.
Thus, the true statements are:
- The distance traveled depends on the amount of time Marlene rides her bike.
- The function [tex]\( f(t) = 16 t \)[/tex] represents the scenario.
