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 ybis the number of miles and x is the time in minutes foe the day you ran the fastest for the week
To find an equation for the number of miles (y) as a function of time (x), we can consider the given information about the two training sessions.
On Monday, you ran 4 miles in 40 minutes. This means your speed was 4 miles / 40 minutes, or 1/10 mile per minute.
On Wednesday, you ran 2 miles in 16 minutes. This implies your speed was 2 miles / 16 minutes, or 1/8 mile per minute.
Since we are looking for the day you ran the fastest, we need to find the highest speed. Comparing the speeds from Monday and Wednesday, we can see that 1/8 is greater than 1/10. Therefore, Wednesday was the day you ran the fastest in terms of speed.
To write an equation, we can assign x as the time in minutes for the day you ran the fastest, which is Wednesday. Likewise, we will assign y as the number of miles you ran.
Hence, the equation for Wednesday's training session is:
y = (1/8)x
To determine the day you ran the fastest during the week, we can compare the average speed for both days. Average speed is calculated by dividing the distance traveled by the time taken.
For Monday, the average speed is 4 miles / 40 minutes = 0.1 miles per minute.
For Wednesday, the average speed is 2 miles / 16 minutes = 0.125 miles per minute.
We can create an equation to find the day with the highest average speed:
0.1x = 0.125y
Here, x is the time in minutes and y is the number of miles. The equation states that the average speed on Monday (0.1 miles per minute) must be equal to the average speed on Wednesday (0.125 miles per minute).
To write an equation for the day you ran the fastest in a week, let's assign the variables:
- Let x represent the time in minutes.
- Let y represent the number of miles covered.
We have two data points from your training:
1. On Monday, you ran 4 miles in 40 minutes: (x1, y1) = (40, 4).
2. On Wednesday, you ran 2 miles in 16 minutes: (x2, y2) = (16, 2).
Now, let's find the equation for the line connecting these two points, which will represent the time and distance at which you ran the fastest:
Step 1: Find the slope (m) using the formula:
m = (y2 - y1) / (x2 - x1)
Substituting the values from the data points:
m = (2 - 4) / (16 - 40)
m = -2 / -24
m = 1/12
Step 2: Use the point-slope form of a linear equation:
y - y1 = m(x - x1)
Choosing (x1, y1) = (40, 4):
y - 4 = (1/12)(x - 40)
Simplifying:
y - 4 = (1/12)x - 40/12
y - 4 = (1/12)x - 10/3
Rearranging the equation, we get:
y = (1/12)x - 10/3 + 4
y = (1/12)x - 10/3 + 12/3
y = (1/12)x + 2/3
Therefore, the equation representing the day you ran the fastest in a week is:
y = (1/12)x + 2/3.