Malia is observing the velocity of a cyclist at different times. After two hours, the velocity of the cyclist is 15 km/h. After five hours, the velocity of the cyclist is 12 km/h.
Part A: Write an equation in two variables in the standard form that can be used to describe the velocity of the cyclist at different times. Show your work and define the variables used. (5 points)
Part B: How can you graph the equations obtained in Part A for the first 12 hours? (5 points)
A. V = Vo + a*t.
V = The velocity in km/h at any time where t is in hours.
a is the acceleration in km/h^2.
Part A:
To write an equation in two variables that describes the velocity of the cyclist at different times, we can use the slope-intercept form of a linear equation: y = mx + b.
Let's define the variables:
- x represents time in hours
- y represents the velocity of the cyclist in km/h
We are given two points:
(2, 15) - velocity after two hours
(5, 12) - velocity after five hours
We can now calculate the slope (m) using the formula: m = (y2 - y1) / (x2 - x1).
m = (12 - 15) / (5 - 2)
m = -3 / 3
m = -1
Now, we can substitute one of the points and the slope into the slope-intercept equation and solve for the y-intercept (b).
Using the point (2, 15):
15 = -1 * 2 + b
15 = -2 + b
b = 17
Therefore, the equation in two variables (time and velocity) is:
y = -x + 17
In standard form, we rearrange the equation:
x + y = 17
So, the equation in standard form is x + y = 17.
Part B:
To graph the equation x + y = 17 for the first 12 hours, we can follow these steps:
1. Set up a coordinate system with the x-axis representing time (in hours) and the y-axis representing velocity (in km/h).
2. Label the x-axis from 0 to 12, representing the first 12 hours.
3. Label the y-axis based on the desired range of velocity. Since the equation does not provide any constraints, you may choose a reasonable range based on the given information and the context of the problem.
4. Plot the point (2, 15) on the graph by locating the x-value of 2 on the x-axis and the y-value of 15 on the y-axis. This point represents the velocity of the cyclist after two hours.
5. Plot the point (5, 12) on the graph by locating the x-value of 5 on the x-axis and the y-value of 12 on the y-axis. This point represents the velocity of the cyclist after five hours.
6. Draw a straight line passing through both points. This line represents the velocity of the cyclist at different times.
7. Extend the line beyond the given points to cover the entire range of the x-axis (0 to 12) if necessary.
8. Label the line as x + y = 17 to indicate the equation it represents.