I need help on making a simple 2 sample word problems of derivative of trigonometric function with solutions

Question

a sample word problem with solution involving derivative of trigonometric function, just can't think a perfect real life situation, i need an opinion from someone. thank you..

Answer 1

Question: A Ferris wheel has a diameter of 50 meters and makes one full rotation in 5 minutes. At what rate is a rider's distance from the ground changing when the rider is 20 meters above the ground?

Solution:

Let θ be the angle (in radians) that the rider's position makes with the ground. Let d be the distance (in meters) that the rider is from the center of the Ferris wheel, and let h be the height (in meters) of the rider above the ground. We can use trigonometry to relate these quantities:

d = (25 meters)cos(θ)

h = (25 meters)sin(θ)

Since the Ferris wheel makes one full rotation in 5 minutes, it completes 2π radians in 5 minutes, or π/150 radians per second. We can use the chain rule to find dh/dt, the rate at which the rider's height above the ground is changing:

dh/dt = dh/dθ * dθ/dt

Taking the derivative of h with respect to θ, we get:

dh/dθ = (25 meters)cos(θ)

Plugging in θ = arccos(4/5) (since cos(θ) = d/25 = 4/5 when the rider is 20 meters above the ground), we get:

dh/dθ = (25 meters)cos(arccos(4/5))

dh/dθ = (25 meters)(4/5)

dh/dθ = 20 meters

Taking the derivative of θ with respect to time, we get:

dθ/dt = π/150 radians per second

Plugging in dh/dθ = 20 meters and dθ/dt = π/150 radians per second, we get:

dh/dt = (20 meters)(π/150 radians per second)

dh/dt = 0.42 meters per second

Therefore, the rider's distance from the ground is changing at a rate of 0.42 meters per second when the rider is 20 meters above the ground.