[填空题]
Annika, Bob, Chloe and Dani go to a fair and take /i9 swe5d k9i*pxyz) rg9kki2a ride on aj)x xx 3w -x+nxf hbh4d)00k(ioki yz5 Ferris wheel. The wheel has nxi35x zhi( fd 0hk)kwxbj x40o)+y-x 8 evenly spaced passenger cars as shown in the following diagram. The wheel completes one rotation in the anticlockwise direction every 80 seconds
Annika starts her ride at position A. Let $h_A$ be Annika's height in meters after t seconds. $h_A$ can be modelled by the function
$h_A$(t)=asin(bt)+d,where a,b,d∈R.
The diagram below shows the graph of y=$h_A$(t) for one revolution.
1.Show that
(1)a=14.
(2)b=$\frac{π}{40}$.
(3)d=18.
At the time Annika starts her ride, Bob is at position B. Bob's position can be modeled by the function $h_B$(t)=14cos($\frac{π}{40}$t)+18.
2.Determine
(1)the first time at which Annika and Bob are at the same height.
(2)the height at which this occurs.
At the time Annika starts her ride, Chloe is at position C. Chloe's position at time t can be written as $h_C$(t)=m(hA(t))+n.
3.Find the value of m and the value of n.
When Chloe reaches an altitude of 3030 meters she has a view of the whole town.
4.(1)Find the time when Chloe has a view of the whole town for the first time.
(2)Find the angle rotated through by Chloe's car from its start position to the point when she has a view of the whole town for the first time. Give your answer in radians, correct to one decimal place.
Dani is at position D when Annika starts her ride. ≈ radians
5.Find the value of c such that the function $h_D$(t)=14sin($\frac{π}{40}$(t−c))+18 describes Dani's height at time t.
The function D(t) represents the difference in height between Annika and Dani's cars. c =
6.Write D(t) as the difference of two sine functions.
D(t) can be written in the form Im($z_1−z_2$), where $z_1$ and $z_2$ are complex functions of t.
7.(1)Write $z_1$ and $z_2$ in exponential form.
(2)Hence or otherwise find an equation for D(t) in the form D(t)=psin(qt+r)+s, where p,q,r,s∈R.
(3)Find the maximum difference in height between Annika and Dani's cars.
