Question

Marion is observing the velocity of a cyclist at different times. After two hours, the velocity is 18km/h. After four hours, the velocity of the cyclist is 4km/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.
Part B: how can you graph the equations obtained in part A for the first 8 hours?

Answer 1

To solve this problem, we assume that velocity is a linear function of time:
v(t) = m t + b
where
v = velocity

t = time

m = slope of line

b = y-intercept of line equation

When t = 2 hours, v = 18 km/h, therefore:
2 m + b = 18 (1)

When t = 4 hours, v = 4 km/h, therefore
4 m + b = 4 (2)

Subtract (1) from (2) to obtain
4 m – 2 m = 4 - 18
2 m = -14
m = -7

From (1), calculate for b:
b = 18 – 2(-7) = 32

Part A. The equation in standard form is therefore:
v = -7 t + 32

Part B. To graph the equation for the first 8 hours, we create a table as shown below. Assign values of t from 0 to 8 hours then calculate the corresponding velocity.

t, hours: 0 1 2 3 4 5 6 7 8
v, km/h: 32 25 18 11 4 -3 -10 -17 -24

From the table, the velocity becomes negative between t=4 and t=5. This means that between this time, Marion already came to rest. Note that when the velocity is zero, t is equivalent to
32 - 7t = 0
7t = 32
t = 4.57

The new table is below and we plot it.
t: 0 1 2 3 4 4.57 5 6 7 8
v: 32 25 18 11 4 0 0 0 0 0