4. Given the table for x, time in minutes, and y, distance traveled:
x: time 8 5 7 2 9
y: distance 664 415 581 166 747
(4 points)
a. Every minute Response area meters are traveled.
b. Write an equation to show the relationship between x and y: Response area
c. This relationship is Response area because the y-intercept is Response area.
a. Every minute, 80 meters are traveled.
b. The equation to show the relationship between x and y is: y = 80x
c. This relationship is linear because the y-intercept is 0.
a. To find the rate at which meters are traveled per minute, we can calculate the difference in distance (y) for each minute (x) value.
For (x=8, y=664), the distance traveled in 8 minutes is 664 meters.
For (x=5, y=415), the distance traveled in 5 minutes is 415 meters.
For (x=7, y=581), the distance traveled in 7 minutes is 581 meters.
For (x=2, y=166), the distance traveled in 2 minutes is 166 meters.
For (x=9, y=747), the distance traveled in 9 minutes is 747 meters.
The differences in distance traveled for each minute are:
8 - 5 = 3 minutes, distance traveled = 664 - 415 = 249 meters
5 - 2 = 3 minutes, distance traveled = 415 - 166 = 249 meters
7 - 5 = 2 minutes, distance traveled = 581 - 415 = 166 meters
2 - 9 = -7 minutes, distance traveled = 166 - 747 = -581 meters
The average rate of distance traveled per minute is the total difference in distance divided by the total difference in time:
(249 + 249 + 166 - 581) / (3 + 3 + 2 - 7) = 83 meters per minute
Therefore, every minute 83 meters are traveled.
b. To write an equation to show the relationship between x and y, we can use the equation of a line in slope-intercept form,
y = mx + b, where m is the slope and b is the y-intercept.
Using the given table of values (x and y), we can find the slope (m) using the formula:
m = (Σxy - nΣxΣy) / (Σx^2 - n(Σx)^2)
n is the number of data points. In this case, n = 5.
Σ denotes sum.
Calculating the required values:
Σxy = (8 * 664) + (5 * 415) + (7 * 581) + (2 * 166) + (9 * 747) = 6517
Σx = 8 + 5 + 7 + 2 + 9 = 31
Σy = 664 + 415 + 581 + 166 + 747 = 2573
Σx^2 = (8^2) + (5^2) + (7^2) + (2^2) + (9^2) = 235
(Σx)^2 = (31)^2 = 961
Plugging in the values:
m = (6517 - (5 * 31 * 2573)) / (235 - (5 * 961))
m = (6517 - 40165) / (235 - 4805)
m = -33648 / -4570
m = 7.366
Now, we can use the y-intercept (b) to complete the equation. From the given table, we can see that when x=0, y is not given. Therefore, we cannot determine the y-intercept (b) from the given information. We would need one more point or additional information to calculate it.
c. As mentioned in part b, we are unable to determine the relationship between x and y because the y-intercept is not given.
a. To find out how many meters are traveled every minute, we need to find the rate at which distance changes with time. We can calculate the difference in distance (y) between consecutive time points (x). Let's do that:
For x = 5, the distance (y) is 415.
For x = 8, the distance (y) is 664.
The difference in distance is 664 - 415 = 249 meters.
So, every minute, 249 meters are traveled.
b. To write an equation showing the relationship between x (time) and y (distance), we can use the slope-intercept form of a linear equation, which is y = mx + b.
Using the given data points, let's find the slope (m) first:
Slope (m) = (change in y) / (change in x)
For x = 2, y = 166
For x = 9, y = 747
(change in y) = 747 - 166 = 581
(change in x) = 9 - 2 = 7
Slope (m) = 581 / 7 = 83
Now, let's find the y-intercept (b) using any of the data points (let's use x = 2 and y = 166):
y = mx + b
166 = 83 * 2 + b
166 = 166 + b
b = 166 - 166
b = 0
Therefore, the equation showing the relationship between x and y is:
y = 83x
c. This relationship is linear because the y-intercept (b) is 0. In a linear equation, if the y-intercept is 0, the equation represents a straight line passing through the origin.