Sinusoidal Word Problems: The Basics

When tackling sinusoidal word problems, we need to identify key values like amplitude (half the distance from maximum to minimum), period (time for one complete cycle), and vertical shift (the middle value).

Let's look at a ferris wheel example: If the highest point is 43 feet high, the wheel has a 40-foot diameter, and completes one rotation every 8 seconds, we can create an equation. The wheel's middle height is 23 feet (highest point minus radius), making our amplitude 20 feet. Since the period is 8 seconds, we use either:

• Using cosine: y = -20cos$π/4·x$ + 23
• Using sine: y = 20sin$π/4·x$ + 23

💡 Remember this pattern: For anything moving in circles (wheels, gears, etc.), the period relates to angular velocity with B = 2π/period, and the amplitude equals the radius!

For objects like bicycle tires with nails, we follow the same process. A 20 cm radius tire making one rotation every 750 ms means the nail's height follows either y = 20cos$8π/3·x$ + 20 or y = 20sin$8π/3(x - 1875)$ + 20. The coefficient 8π/3 comes from converting the period to radians $2π/0.75$.

Solving More Complex Motion Problems

Roller coaster heights and bouncing springs also follow sinusoidal patterns. For these problems, we need to determine the middle point (vertical shift), the amplitude, and the period from the given information.

For the roller coaster problem, if John reaches a maximum height of 12 m at 13.2 seconds and a minimum height of 4 m at 14.6 seconds, we can determine:

• Vertical shift: 8 m (average of max and min)
• Amplitude: 4 m (half the difference between max and min)
• Period: 2.8 seconds (twice the time between max and min)

This gives us equations like y = -4cos$5π/7(x-13.2)$ + 8 or y = -4sin$5π/7·x$ + 8.

🔑 When you have two known points on a sinusoidal graph, find the period by determining how long it takes to go from max to min and then doubling it!

For a bouncing spring that reaches 60 cm at 0.3 seconds and 40 cm at 1.9 seconds, we identify the vertical shift as 50 cm and amplitude as 10 cm. The period is 3.2 seconds (twice the time from high to low point). Using 2π/period gives us the coefficient for our equations.