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
meters are traveled.
b. Write an equation to show the relationship between x and y:
.
c. This relationship is
because the y-intercept is
.
a. To find the rate at which meters are traveled per minute, we can calculate the average distance traveled per minute. The total distance traveled is:
664 + 415 + 581 + 166 + 747 = 2,573
The total time taken is:
8 + 5 + 7 + 2 + 9 = 31
Therefore, the average distance traveled per minute is:
2,573 / 31 ≈ 83.06
So, approximately 83 meters are traveled every minute.
b. To write an equation to show the relationship between x (time) and y (distance), we can use a linear equation in the form y = mx + c, where m is the slope and c is the y-intercept.
We can find the slope by finding the change in y divided by the change in x. Taking two points, let's say (8, 664) and (2, 166), the change in y is:
664 - 166 = 498
And the change in x is:
8 - 2 = 6
So, the slope is:
m = 498 / 6 = 83
Therefore, the equation is:
y = 83x + c
To find the value of c, we can substitute any of the given points into the equation. Let's use the point (8, 664):
664 = 83(8) + c
Simplifying:
664 = 664 + c
c = 0
So, the equation is:
y = 83x
c. This relationship is linear because the equation is in the form of a linear equation: y = 83x. The y-intercept is 0.
To find the answers, we need to analyze the given table.
a. To calculate the distance traveled every minute, we need to find the difference in distance traveled between consecutive minutes. We can do this by subtracting the distance at the starting minute from the distance at the ending minute and then dividing by the number of minutes.
So, the distance traveled every minute is:
(415 - 664) / (5 - 8) = -249 / -3 = 83 meters
Therefore, every minute, 83 meters are traveled.
b. To write an equation to show the relationship between x (time) and y (distance traveled), we can use the method of interpolation. One equation that fits the given data points is a linear equation of the form y = mx + c, where m is the slope and c is the y-intercept.
To find the slope, we can use any two data points. Let's use the points (2, 166) and (9, 747).
m = (747 - 166) / (9 - 2)
= 581 / 7
= 83 meters/minute
Now, to find the y-intercept (c), we can substitute one of the points into the equation and solve for c. Let's use the point (5, 415).
415 = 83 * 5 + c
415 = 415 + c
c = 0
Therefore, the equation that represents the relationship between x and y is:
y = 83x
c. This relationship is a linear relationship because the equation y = 83x is in the form of a linear equation, where the power of x is 1 (i.e., x is raised to the power of 1).
The y-intercept is 0 because when x = 0, the equation y = 83x simplifies to y = 83 * 0, which is equal to 0.
To answer the given questions, we need to analyze the data provided in the table. Let's go step by step:
a. To find how many meters are traveled every minute, we need to calculate the rate of change of distance with respect to time. We can do this by comparing the distances at two different time points. Let's take the first and second time points, x = 8 and x = 5, and their corresponding distances y = 664 and y = 415, respectively.
Change in distance = Final distance - Initial distance = 415 - 664 = -249
Change in time = Final time - Initial time = 5 - 8 = -3
Now, we can find the rate of change per minute:
Rate of change = Change in distance / Change in time = -249 / -3 = 83
Therefore, every minute, 83 meters are traveled.
b. To write an equation showing the relationship between x (time) and y (distance), we can use the concept of a linear equation. A linear equation can be written in the form y = mx + b, where m is the slope (rate of change) and b is the y-intercept.
Using the data from the table, we can choose any two points. Let's take x = 2 and y = 166 as one point and x = 9 and y = 747 as another point.
Slope (m) = (y2 - y1) / (x2 - x1) = (747 - 166) / (9 - 2) = 581/7
Now, substituting any of the two points and the slope in the equation y = mx + b, we can solve for b:
166 = (581/7) * 2 + b
Simplifying the equation:
166 = 1166/7 + b
166 - 1166/7 = b
b = 1166/7 - 1166/7
b = 0
Therefore, the equation showing the relationship between x and y is: y = (581/7)x.
c. To determine the nature of the relationship, we need to consider the y-intercept (b) in the equation y = mx + b.
From the previous calculation, we found that the y-intercept is b = 0.
Since the y-intercept is zero, it implies that when x (time) is equal to zero, the distance traveled (y) is also zero. This means the object starts at the origin (0,0) and there is a linear relationship between time and distance. So, the relationship is linear, and the y-intercept is 0.