Week 5: Double Modelling Challenge

🎡 The Scenario

Manu and Jade are at the "Maths End" amusement park. They go on two different Ferris wheels at the same time ($t=0$).

1. The Kiddy-Wheel (Jade):

Max Height: 8 m. Loading Ramp: 0.5 m.
Speed: 2 revolutions per minute.
Jade starts at the bottom ($t=0$).

2. The Flying-High (Manu):

Max Height: 43 m. Min Height: 3 m.
Speed: 3 revolutions in 2 minutes.
Manu boards at the mid-point, moving upwards ($t=0$).

The Problem: Due to trees, Jade can only see Manu when:

Jade is higher than 5 m.
Manu is higher than 25 m.

🏆 Marking Schedule

Achievement: Form a trigonometric model for ONE wheel (Jade OR Manu). Define variables correctly.

Merit: Form correct models for BOTH wheels and use them to find a simple value (e.g., height at $t=10$).

Excellence: Form a generalisation or chain of reasoning. Find the time intervals where they can see each other (solving simultaneous inequalities).

Step 1: Jade's Model (Negative Cosine)

Since Jade starts at the bottom, a $- \cos$ curve is the smartest choice (no horizontal shift $C$).

Form: $h_J(t) = -A \cos(Bt) + D$

1. Vertical Shift (D): $\frac{Max + Min}{2}$

2. Amplitude (A): $Max - D$

3. Frequency (B): 2 revs/min $\rightarrow$ Period = 30s.

Visual Check

Enter: $y = -3.75\cos(0.209x) + 4.25$

Step 2: Manu's Model (Sine)

Manu starts at the mid-point going up. This is exactly what a $+\sin$ curve does!

Form: $h_M(t) = A \sin(Bt) + D$

1. Vertical Shift (D): $\frac{43 + 3}{2}$

2. Amplitude (A): $43 - 23$

3. Frequency (B): 3 revs in 120s $\rightarrow$ Period = 40s.

Visual Check

Enter: $y = 20\sin(0.157x) + 23$

🚀 Excellence Criteria

To get E7/E8, you must:

State constraints clearly: $h_J > 5$ AND $h_M > 25$.
Graph the intersection: Show the regions of overlap.
Calculate Interval: State the start and end times clearly (e.g., $t_1 < t < t_2$).
Long-Term Trend: Explain how often this interval repeats.

Step 1: Interactive Solver

Instructions:

Input Jade's Model: $y = -3.75\cos(0.209x) + 4.25$
Input Manu's Model: $y = 20\sin(0.157x) + 23$
Input Constraint 1: $y > 5$ (Shade Green)
Input Constraint 2: $y > 25$ (Shade Purple)

Note: The graph shows the first 130 seconds so you can see the repeating pattern.

Task: Find ANY time interval (in seconds) where they can see each other.

Step 2: Long-Term Trend Analysis

Why are there multiple answers?

Jade's Cycle = 30s. Manu's Cycle = 40s.
They drift in and out of sync multiple times!
Common Period (LCM): They reset perfectly every 120 seconds.

The "Pattern" of Visibility:

Within every 120s cycle, there are four opportunities:

$8.5 < t < 19.4$ (The obvious one)
$40.6 < t < 51.5$ (The one you found!)
$80.6 < t < 81.5$ (A tiny overlap)
$98.5 < t < 99.4$ (Another short one)

Conclusion: To answer the "Long Term Trend" question, you pick one of these and say it repeats every $120k$ seconds.