You are training twice a week for a race. On monday, you go 4 miles in 40 minutes. then on wednesday you go 2 miles in 16 minutes. write an equation where y is the number of miles and x is the time in minutes for the day you ran the fastest for the week.
Let's assume that the equation represents a line, where y represents the number of miles and x represents the time in minutes. The equation for this line can be written in the form:
y = mx + b
To find the equation, we need to determine the slope (m) and the y-intercept (b) of the line.
First, let's determine the rate of speed for each run:
For Monday:
Distance (y1) = 4 miles
Time (x1) = 40 minutes
Rate of Speed (m1) = y1/x1 = 4/40 = 1/10 mile per minute
For Wednesday:
Distance (y2) = 2 miles
Time (x2) = 16 minutes
Rate of Speed (m2) = y2/x2 = 2/16 = 1/8 mile per minute
Since we are looking for the day you ran the fastest, we need to find the higher rate of speed. Let's compare the values:
m1 = 1/10 mile per minute
m2 = 1/8 mile per minute
The higher rate of speed is 1/8 mile per minute from Wednesday's run, so we will use m = 1/8.
Now let's find the y-intercept (b). We can use the point-slope form of the equation:
y - y1 = m(x - x1)
Using the coordinates from Wednesday's run (2, 16):
y - 2 = (1/8)(x - 16)
y - 2 = (1/8)x - 2
y = (1/8)x
Thus, the equation representing the day you ran the fastest is:
y = (1/8)x
To write an equation representing the relationship between distance (y) and time (x) for the fastest day of the week, we need to compare the times it took to run on Monday and Wednesday.
On Monday, you ran 4 miles in 40 minutes, giving us the equation:
40 = 4y (since the rate is constant)
On Wednesday, you ran 2 miles in 16 minutes, giving us the equation:
16 = 2y
To find the fastest day, we need to compare the values of y for both equations. To simplify the equations, we can divide both sides by the distance (y):
40/4 = y
10 = y
16/2 = y
8 = y
From these equations, we can see that the fastest day was Wednesday, where y (distance) was 8 miles and x (time) was 16 minutes. The equation representing this day is:
16 = 8y
To find the day you ran the fastest for the week, we need to compare the pace or speed at which you ran on both Monday and Wednesday. The equation we can use to determine this is:
y = x/t
Where:
- y represents the number of miles
- x represents the time in minutes
- t represents the speed or pace in minutes per mile
Let's calculate the pace for Monday first. You ran 4 miles in 40 minutes, so the pace on Monday can be found by dividing the time (x) by the distance (y):
t1 = x1/y1
t1 = 40/4
t1 = 10 minutes per mile
Now, let's calculate the pace for Wednesday. You ran 2 miles in 16 minutes, so the pace on Wednesday can be found by dividing the time (x) by the distance (y):
t2 = x2/y2
t2 = 16/2
t2 = 8 minutes per mile
Comparing the two pace values, we can see that Wednesday had the faster pace since it took you only 8 minutes per mile, while Monday took 10 minutes per mile. Therefore, the equation for the day you ran the fastest is:
y = x/8