Thread: Application of trig

The depth, D(t) metres, of water at the entrance to a harbour at t hours after midnight on a
particular day is given by D(t) = 10 + 3 sin(π t/6), 0 ≤ t ≤ 24.
a) Find the value of t for which D(t) ≥ 8.5.
b) Boats which need a depth of w metres are permitted to enter the harbour only if the
depth of the water at the entrance is at least w metres for a continuous period of 1 hour.
Find, correct to 1 decimal place, the largest value of w which satisfies this condition.

i was able to sketch the graph, just having trouble with how you figure out D(t) ≥ 8.5

the answer in the book is {t : D(t) ≥ 8.5} = {t : 0 ≤ t ≤ 7} ∪
{t : 11 ≤ t ≤ 19} ∪ {t : 23 ≤ t ≤ 24}
this makes sense, i just dont know how to figure it out algebraically

Hello needhelpplease

Originally Posted by needhelpplease

The depth, D(t) metres, of water at the entrance to a harbour at t hours after midnight on a
particular day is given by D(t) = 10 + 3 sin(π t/6), 0 ≤ t ≤ 24.
a) Find the value of t for which D(t) ≥ 8.5.
b) Boats which need a depth of w metres are permitted to enter the harbour only if the
depth of the water at the entrance is at least w metres for a continuous period of 1 hour.
Find, correct to 1 decimal place, the largest value of w which satisfies this condition.

i was able to sketch the graph, just having trouble with how you figure out D(t) ≥ 8.5

the answer in the book is {t : D(t) ≥ 8.5} = {t : 0 ≤ t ≤ 7} ∪
{t : 11 ≤ t ≤ 19} ∪ {t : 23 ≤ t ≤ 24}
this makes sense, i just dont know how to figure it out algebraically

Here's my quick sketch of the graph (see attachment). Clearly when , and we can find the values for which by solving the equation .

So:









(Is this where you were getting stuck? Look at the point moving around the unit circle, noting that . So the first value of for which is , etc)



... and using the sketch-graph, the answers then follow.

Grandad

thanks that really helps
i was getting to here sin(π t/6)=-0.5 and then solving for t, didn't realise you had to use the unit circle.
makes sense now, thanks