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Periodic Trig Function Models - Word Problems

The following are word problems that use periodic trigonometry functions to model behavior.

Solutions are in the images below.

1) A ferris wheel is 4 feet off the ground. It has a diameter of 26 feet, and rotates once every 32 seconds. If you begin the ride sitting in a chair that is 6 feet above the ground, how high will you be 10 seconds into the ride? During the first minute, when will you be 20 feet high?

2) A car's tire has a diameter of 32 inches. It runs over a nail, but it is able to continue moving. Write a cosine function that describes the height of the nail above ground as a function of angular distance.

3) Each day, the tide continuously goes in and out, raising and lowering a boat (sinuisoidally) in the harbor. At low tide, the boat is only 2 feet above the ocean floor. And, 6 hours later, at peak high tide, the boat is 40 feet above the ocean floor. Write a sine function that describes the boat's distance above the ocean floor as it relates to time.

For safety, the boat needs 14 feet of depth to sail. If high tide occurs at noon, between what times can the boat go out to sea?

4) The following trig function models the position of a rung on a waterwheel:

where t = seconds and y = number of feet above water level.

a) What is the diameter of the wheel?

b) At the top of the wheel, how high is the rung above water level?

c) How many rotations per minute does the wheel make?

d) What percentage of time does a rung spend underwater?

5) The motion of a swing hanging from a tree next to a lake can be modeled by a sinusoid. The tree is 10 feet from the water, and the swing can extend 20 feet from the tree in each direction. If it takes 2 seconds to swing from one side to the other side,

a) write an equation that models the position of the swing as a function of time.

b) determine the interval of time that the swing is above the water.

c) what percentage of time is the swing above the water?

6) A clock with a 14 inch diameter has a minute hand that is 6" long and an hour hand that is 4" in length. Assume a line drawn from the 9 to the 3 represents an elevation of zero.

a) Write a sine model showing the position (elevation) of the minute hand tip as it relates to time.

What is the elevation of the minute hand tip at 53 minutes past the hour?

When is the elevation of the minute hand tip 2 inches below the elevation zero line?