The distance remaining for a half marathon race over several minutes is shown in table. Use the information to determine the constant rate change in the minutes per mile.
Table:
Time (min) Distance Remaining (mi)
40 8
56 6
72 4
88 2
If the rate is constant, then
t = md+b
Now plug in two pairs of values, say the first two.
8 = 40a + b
6 = 56a + b
So, a = -1/8 and b=13
t = -1/8 d + 13
Now check the other two pairs of values to make sure they agree.
time = (distance-13)/-.125
ex. (8 + -13)/-.125 = 40
(6 + -13)/-.125 = 56
To determine the constant rate of change in minutes per mile, we need to calculate the change in distance divided by the change in time between two points on the table.
The change in distance between 40 minutes (8 miles) and 88 minutes (2 miles) is 8 - 2 = 6 miles.
The change in time between 40 minutes and 88 minutes is 88 - 40 = 48 minutes.
Therefore, the constant rate of change in minutes per mile can be calculated by dividing the change in time by the change in distance:
48 minutes / 6 miles = 8 minutes per mile.
Hence, the constant rate of change in minutes per mile is 8 minutes per mile.
To determine the constant rate change in minutes per mile, we need to find the change in distance per minute.
Let's analyze the given information with the table:
The initial distance remaining is 8 miles, and after 40 minutes, the distance remaining is 6 miles. This means that in 40 - 0 = 40 minutes, the distance decreased by 8 - 6 = 2 miles.
Therefore, the rate of change is 2 miles in 40 minutes, or 2/40 = 0.05 miles per minute.
Similarly, after another 16 minutes (56 - 40), the distance decreased by 6 - 4 = 2 miles.
Hence, the rate of change is 2 miles in 16 minutes, or 2/16 = 0.125 miles per minute.
Finally, after another 16 minutes (72 - 56), the distance decreased by 4 - 2 = 2 miles.
So, the rate of change is 2 miles in 16 minutes, or 2/16 = 0.125 miles per minute.
Since the rate of change in all three cases is the same, we can conclude that the constant rate change in minutes per mile is 0.125 minutes per mile.