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6) A platform which starts fifteen metres above the ground & goes up & down.
The distance = s metres of the platform above the ground, t = seconds after the ride starts, can be modelled by the function: $s(t)=15+6sin(\pi/3)$

a)

(i) According to this model, find the maximum height above the ground reached by the platform.

(ii) Show that the platform completes 1 cycle in 6secs.

(iii) How many secs after the ride starts is it 9m above the ground for the 1st time ?

(iv) Sketch a graph for $0≤ t ≤60$ secs.

(v) How high above the ground is it when $t = 52.5$s?

They want the platform to go further up & down, during the first 60 secs of the new platform, the distance r metres, of the platform above the ground, t seconds after the ride starts, is modelled by the function $r(t)= 15+ e^{0.04t} sin(\pi t/3)$ for $0≤ t ≤60$

(b)

(i) Explain why the platforms period is still 6secs.

(ii) Explain in general terms why it takes longer to reach the max height of the first ride but will exceed it.

(iii) How high above the ground is it when $t = 4.5$s and $t = 52.5$s ?

(iv) After how many secs <60 does it reach the biggest height above ground, what is that height ?

(v) After how many secs <60 does it reach its lowest height above ground, what is that height ?

(vi) Sketch a graph of the ride for $0≤ t ≤60$ seconds.

They changed the platform according to the model $f(t)=15+e^{0.05t} sin(\pi t/3)$ for $0<t<60$

(c) Explain why this would have been catastrophic.

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This function doesn't have the variable $t$ in it, are you sure it is the right function: $s(t)=15+6\sin\left(\dfrac{\pi}{3}\right)$ –  Hans Groeffen Mar 3 '13 at 15:07
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1 Answer

Hints: for a) the sine is between $-1$ and $1$. Which gives the maximum? for b) you have to increase the argument of the sine function by $2\pi$ for one period. It looks like the function should be $15+6\sin (\frac {\pi t}3)$ and you lost the $t$. How much increase in $t$ is needed for that? for c)solve the position equation $9=15+6\sin(\frac {\pi t}3)$ for $\sin(\frac {\pi t}3)$, then find the minimum $t$ that satisfies that.

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