If the pendulum swings out to its widest position in 2 seconds, model the height of the pendulum from the ground using a cosine function, considering vertical to be t = 0.

Question

A pendulum is connected to a rope 3 m long, which is connected to a ceiling 4 m high. The angle between its widest swing and vertical hanging position is . If the pendulum swings out to its widest position in 2 seconds, model the height of the pendulum from the ground using a cosine function, considering vertical to be t = 0.

Grade 12 Advanced Functions

Please write neatly and clearly. Thanks

Answer 1

Hi!

It is first easier to calculate the distance from the ceiling since the trigonometric relations are then obvious.

The vertical distance to the ceiling is just

3m·cos(φ). Since the highest angle is just φ = π/3 and cos(π/3) = 1/2, the smallest distance to the ceiling is just 1.5 meters and the highest point is 2.5 meters from the ground. This happens after 2s, thus φ = π/6·t Thus the vertical distance to the ceiling should be modeled as:

3·cos(π/6t)

The distance to the ground is just 4 minus the distance to the ceiling, this means:

h(t) = 4-3cos(π/6t).

We check, that h(0) = 1 and h(t=2s) = 2.5

Hope, this helps