A group of people are tubing down a river at an average speed of mi/h. Write a direct variation equation that gives the number of miles y that the people will float in x hours. Graph the equation and use the graph to find the distance traveled in 2.5 hours.
To write a direct variation equation, we can use the equation form y = kx, where y represents the number of miles floated in x hours, and k is the constant of variation.
From the given information that the group is tubing down the river at an average speed of mi/h, we can say that the number of miles floated is directly proportional to the time taken.
Hence, the direct variation equation is:
y = kx
Now, to find the value of k, we can use the given information that the average speed is mi/h. This means that the distance traveled per hour is mi.
So, substituting the values of y = mi and x = 1 hour into the direct variation equation, we get:
mi = k(1)
Simplifying,
k = mi/h
Therefore, the direct variation equation is:
y = (mi/h)x
To graph the equation, we can plot the points for different values of x and y. Let's choose x = 0, 1, 2, 3, and so on, and find the corresponding values of y.
For x = 0, y = (mi/h) * 0 = 0 miles floated
For x = 1, y = (mi/h) * 1 = mi floated
For x = 2, y = (mi/h) * 2 = 2mi floated
For x = 3, y = (mi/h) * 3 = 3mi floated
And so on...
Now, let's plot these points on a graph and draw a line through them.
To find the distance traveled in 2.5 hours, we can look at the graph and find the corresponding value of y when x = 2.5.
The distance traveled is approximately the value of y on the graph when x = 2.5.
Please note that without specific values for the average speed, the units used, or more data points, we cannot provide precise graph coordinates or the exact distance traveled in 2.5 hours.
To write a direct variation equation, we need to determine the constant of variation. In this case, the average speed of tubing down the river, in miles per hour (mi/h), is the constant of variation.
Let's denote the constant of variation as k. Since the number of miles floated, y, is directly proportional to the number of hours, x, the direct variation equation is:
y = kx
We know that the average speed of tubing down the river is mi/h, which means that k = mi/h. Therefore, the direct variation equation becomes:
y = (mi/h) * x
Now, let's use this equation to find the distance traveled in 2.5 hours.
To find the distance traveled in 2.5 hours, we substitute x = 2.5 into the equation:
y = (mi/h) * 2.5
Simplifying this equation, we have:
y = 2.5 * (mi/h)
Hence, the distance traveled in 2.5 hours is 2.5 * (mi/h) miles.
To graph the equation, we can choose any arbitrary value for mi/h, let's say mi/h = 2. The equation becomes:
y = 2.5 * 2
Simplifying,
y = 5
This means that for every 1 hour, the group of people will float 5 miles.
Now, let's plot this equation on a graph.
The x-axis represents time (in hours) and the y-axis represents the distance traveled (in miles). We can plot a few points on the graph:
(1, 5)
(2, 10)
(3, 15)
(4, 20)
Connecting these points on the graph, we obtain a straight line passing through the origin. This line represents the direct variation equation y = 2.5x.
To find the distance traveled in 2.5 hours, we find the corresponding y-coordinate on the graph, which is y = 2.5 * 2.5 = 6.25 miles.
Therefore, the distance traveled in 2.5 hours is 6.25 miles.
please proofread
y = s x
straight line through origin of slope s, that mysterious speed