Final answer:
Jimmy's independent variable is time, and the dependent variable is the distance from home. His rate of change (speed) is 93 km/h, with the initial value being 1019km from home. After 8h and 15m, he will be 243.75km from home, and it will take him approximately 11 hours to get home.
Step-by-step explanation:
Let's analyze Jimmy's trip and answer the questions one by one:
a. Independent and Dependent Variables
The independent variable is the time in hours, since it is not dependent on another variable but is instead a control variable in this scenario. The dependent variable is the distance from home in kilometers, because it changes in response to the time spent driving.
b. Rate of Change
The rate of change represents the speed at which Jimmy is driving. To calculate it, we subtract the final distance from the initial distance and divide by the time interval: (461km - 740km) / (6h - 3h) = -279km / 3h = -93 km/h. The negative sign indicates he is moving towards home.
c. Initial Value
The initial value is the distance from home at the start of the observation period. Here, after 3 hours of driving, Jimmy was 740km from home. Thus, the initial value is 740km plus the distance covered in the first 3 hours. To find this, we add the product of the rate of change and the time: 740km + (93 km/h * 3h) = 1019km. So, Jimmy was 1019km from home when he started his trip.
d. Distance After 8h and 15m
To calculate the distance after 8 hours and 15 minutes (8.25h), we use the formula: Distance = Initial Value + (Rate of Change * Time). This gives us 1019km - (93 km/h * 8.25h) which equals 243.75km from home.
e. Time to Get Home
To find out the time it will take him to get home, we set the final distance to zero and solve for time: 0 = 1019 km - 93 km/h * t. This results in t = 1019 km / 93 km/h, which is approximately 10.96 hours or roughly 11 hours.