Final answer:
The relationship between time and distance for Justine and her family floating down a river is linear, as the rate of distance covered per hour is constant at 1.25 miles/hour. Therefore, the function modeling this relationship is y = 1.25x, where x is time in hours and y is the distance in miles.
Step-by-step explanation:
To determine if the relationship between time and distance is linear for Justine and her family floating down the river, we can look at the ratios of distance to time at the given points. For a linear relationship, the ratio of distance traveled to time should be constant.
After 1 hour, the ratio is 1.25 miles to 1 hour, which is 1.25 miles/hour. After 4 hours, the ratio is 5 miles to 4 hours, which is also 1.25 miles/hour. After 6 hours, they have floated 7.5 miles, with a ratio of 7.5 miles to 6 hours, or again 1.25 miles/hour. Since the ratio is consistent, the relationship is linear.
To write a function that models this linear relationship, we use the form y = mx + b, where y is the distance, x is the time, m is the slope (rate of change in distance per hour), and b is the y-intercept. As there is no distance covered at time zero (assuming they start at the origin), the y-intercept b is 0.
Thus, the function is:
y = 1.25x
Where x is the number of hours and y is the distance in miles.