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**Macleef** **A particle moves along a line. The particle's position, $\displaystyle s$, in meters at $\displaystyle t$ seconds is modelled by $\displaystyle s(t) = 2t^{3} - 15t^2 + 36t + 40$, where $\displaystyle t \geq 0$.**

**e) Determine the total distance travelled during the first five seconds**

Found the first derivative, used interval chart, found distances for time found and tried to find the total distance for the first 5 seconds:

$\displaystyle 0 = 6t^2 - 30t + 36$

$\displaystyle 0 = 6(t - 3)(t - 2)$

$\displaystyle t = 2s $ and $\displaystyle t = 3s$

$\displaystyle (0, 2) \rightarrow$ advancing

$\displaystyle (2, 3) \rightarrow$ retreating

$\displaystyle (3, \infty) \rightarrow$ advancing

$\displaystyle s(0) = 40m$

$\displaystyle s(2) = 68m$

$\displaystyle s(3) = 67m$

$\displaystyle s(5) = 95m$

$\displaystyle s(0) = 40m$

$\displaystyle s(3) - s(2) = 1m$

$\displaystyle s(5) - s(3) = 28m$

$\displaystyle 40m + 1m + 28m = 69m$

**Textbook Answer: 57m**