# sine graph

• Mar 23rd 2006, 02:07 AM
rachael
Carnival Ride (was: exponential)
8 The height above the ground, x metres, of a person on a ride at a carnival at time t seconds after the ride started moving can be modelled by the function x(t) = 6 cos [pi[t]\24] + 6
a How long, in seconds, does the ride take to complete a revolution?
b What is the exact height of the ride when t = 20?
c Write down the minimum and maximum heights reached in metres.
d Sketch the graph of x versus t, 0 ≤ t ≤ 120.
e For what values of t, during the first 60 seconds of the ride, is the height at most 3 metres?

i need help on part e. how do i work out this?
• Mar 23rd 2006, 02:40 AM
CaptainBlack
Quote:

Originally Posted by rachael
8 The height above the ground, x metres, of a person on a ride at a carnival at time t seconds after the ride started moving can be modelled by the function x(t) = 6 cos [pi[t]\24] + 6
a How long, in seconds, does the ride take to complete a revolution?
b What is the exact height of the ride when t = 20?
c Write down the minimum and maximum heights reached in metres.
d Sketch the graph of x versus t, 0 ≤ t ≤ 120.
e For what values of t, during the first 60 seconds of the ride, is the height at most 3 metres?

i need help on part e. how do i work out this?

Part e. is asking for what values of t in the first 60 seconds is x(t)<=3m.

From your sketch of the graph of x(t) in part d you will know that
the range of t satisfying this condition is ~15-30 seconds. Further
checking will show that to find the exact end points of this interval
we need the roots of:

$
x(t) = 6 \cos (\pi\ t/24) + 6=3
$

which may be rearranged to:

$
\cos (\pi\ t/24)=-1/2
$

Now the angles which have a cosine of -1/2 should be well known to
you, these are:

$\theta=\pi\ t/24=\pm \frac{2}{3}\pi+2n\pi,\ \ n=0, \pm 1, \dots$.

Then with a little bit of experiment you will find that the times close to 15
and 30 seconds which are solutions to this are 16 and 32 seconds.

So we have the result for t in the closed interval [16,32] seconds the height is at most 3m.

RonL