TRIG 12 DR WHO AND THE DALEK
Dr Who and the Dalek are going to go on two different Ferris wheels. One wheel is smaller than the other. Dr Who accesses the smaller wheel by a ramp which is 2.5 m off the ground. The wheel reaches a maximum height of 21 m. The Dalek accesses a larger Ferris wheel by a ramp that is 3.5 m above ground and reaches a maximum height of 38 m above ground. When in range The Dalek is capable of exterminating Dr Who. At other times on the ride he cannot. Both rides last 2 minutes.
They both got on their Ferris wheels at the lowest point and at the same time. Dr Who's Ferris wheel is going around 2 revolutions/minute and The Dalek's, 3 revolutions in 2 minutes. Sketch a diagram showing the wave forms for both graphs verified with your calculator. Find the times at which Dr Who is within range of The Dalek. The Dalek attacks Dr Who if he is ascending (rising) and is more than 10 m off the ground but less than 16 m off the ground.
AMPLITUDE A
Doctor: (max - min) /2 = (21 - 2.5)/2 = 9.25 m
Dalek: (max - min) /2 = (38 - 3.5)/2 = 17.25 m
Dalek: (max - min) /2 = (38 - 3.5)/2 = 17.25 m
FREQUENCY B
The frequency B is equal to the distance of 2π divided by the time for one period.
Doctor's wheel , is: 2π/30 = π/15 so B = π/15
Dalek's wheel : is 2π/40 = π/20 so B = π/20
Doctor's wheel , is: 2π/30 = π/15 so B = π/15
Dalek's wheel : is 2π/40 = π/20 so B = π/20
HORIZONTAL SHIFT C
Shift = - (time to maximum - 1/4 of period) = -(15 - 7.5) = -7.5
For The Dr, this value is 15 seconds (the time to get to max height) take away 7.5 seconds, (1/4 x 30).
So the value of C = -7.5 and is written as (t - 7.5) in the equation
For The Dalek, this value is 10 seconds (the time to get to max height) take away 10 seconds, (1/4 x 40)
So the value of C = -10 seconds and is written as (t - 10) in the equation
For The Dr, this value is 15 seconds (the time to get to max height) take away 7.5 seconds, (1/4 x 30).
So the value of C = -7.5 and is written as (t - 7.5) in the equation
For The Dalek, this value is 10 seconds (the time to get to max height) take away 10 seconds, (1/4 x 40)
So the value of C = -10 seconds and is written as (t - 10) in the equation
VERTICAL SHIFT D
Both Ferris wheels are at different heights,
The Dr's wheel = (max + min )/2 = (21 + 2.5)/2 = 11.75 m above the ground, so D = 11.75
The Dalek's wheel = (max + min )/2 = (38 + 3.5)/2 = 20.75 m above the ground, so D = 20.75
The Dr's wheel = (max + min )/2 = (21 + 2.5)/2 = 11.75 m above the ground, so D = 11.75
The Dalek's wheel = (max + min )/2 = (38 + 3.5)/2 = 20.75 m above the ground, so D = 20.75
THE EQUATIONS
The Dr: h(t) = 9.25sin(π/15(t - 7.5)) + 11.75
The Dalek: h(t) = 17.25sin(π/20(t - 10)) + 20.75
The Dalek: h(t) = 17.25sin(π/20(t - 10)) + 20.75
WAVEFORM SKETCHES - THE SMALL WHEEL
For the small wheel you only need 3 points solved on the graph to get Achieved, and the equation of course.