Question

Simon travels north and south from Main Station. The distance, in km, of the train from
Main Station is modeled by the function d(t)=t^3-9t^2+6t, where North is positive and
South is negative. Time elapsed after the start of a shift, in hours, is represented by t,
where 𝑡 ∈ [0,12]. If the shift starts at noon, determine at which time(s) the train is more
than 16 km south of Main Station.

Answer 1

you just want

d(t) > -16

t^3-9t^2+6t > -16
t^3-9t^2+6t + 16 > 0
(t+1)(t-2)(t-8) > 0

Now, using what you know about the general shape of cubics, you know that this one will come up from the lower left, then cross the t-axis at -1,2,8. Since it started out negative, it will be positive on the intervals

(-1,2) and (8,∞)

Since our domain is [0,12], modify those intervals to fit the domain.

Check your answers against the graph at

http://www.wolframalpha.com/input/?i=t%5E3-9t%5E2%2B6t+%3E+-16