A runner runs a 10 metre race, the runner runs according to the following rule:

s = 0.2t(t - 5.1)(t - 9.1)
Where displacement (s) is in metres and time (t) is in minutes.

Calculate for the runner:
their starting position
any changes in direction over the race (when and where it happened)
their finishing time for the race

s(0) = 0

a little calculus (or inspection of the graph) will show that the runner changed direction at t=2.1 and t=7.4

s=10 when t≈10

http://www.wolframalpha.com/input/?i=plot+y%3D0.2t%28t+-+5.1%29%28t+-+9.1%29%2C+y%3D10

To calculate the runner's starting position, we need to substitute t=0 into the equation for displacement (s).

s = 0.2t(t - 5.1)(t - 9.1)
s = 0.2(0)(-5.1)(-9.1)
s = 0

Therefore, the runner's starting position is at 0 meters.

Now let's find any changes in direction over the race. Changes in direction occur when the velocity changes from positive to negative or vice versa. We can determine this by examining the sign of the velocity.

To find the velocity, we need to take the derivative of the displacement equation with respect to time (t):

v = ds/dt = d/dt [0.2t(t - 5.1)(t - 9.1)]

To make it easier, let's simplify the equation first:

s = 0.2t(t^2 - 5.1t - 9.1t + 46.41)
s = 0.2t(t^2 - 14.2t + 46.41)

Now differentiate it:

v = 0.2 [3t^2 - 14.2t + 46.41] + [0.2t(2t - 14.2) + 0.2t(t - 5.1)]
v = 0.2 [3t^2 - 14.2t + 46.41 + 4t^2 - 28.4t + t^2 - 5.1t]
v = 0.2 [8t^2 - 47.7t + 46.41]

Now we can find the values of t where the velocity changes its sign:

0.2 [8t^2 - 47.7t + 46.41]=0

After simplifying and solving the quadratic equation, we find that the velocity changes sign at t = 2.67 minutes and t = 7.43 minutes.

To determine where the change in direction occurs, we can substitute these values back into the displacement equation:

s(2.67) = 0.2(2.67)(2.67 - 5.1)(2.67 - 9.1)
s(7.43) = 0.2(7.43)(7.43 - 5.1)(7.43 - 9.1)

Solving these equations will give us the exact positions where the changes in direction occur during the race.

Lastly, to find the finishing time for the race, we need to set the displacement equation equal to 10 (since it's a 10m race) and solve for t:

10 = 0.2t(t - 5.1)(t - 9.1)

Again, solving this equation will give us the finishing time for the runner in minutes.