Applications of Sinusoidal Functions

Application of Sinusoidal Functions

Pump Jack Problem

A pumpjack is the overground drive for a reciprocating piston pump in an oil well. It is used to mechanically lift liquid out of the well if not enough bottom hole pressure exists for the liquid to flow all the way to the surface.

The head of a pump jack moves up and down. As it does so, the distance the head of the pump jack is from the ground varies sinusoidally with time.

Distance, d in meters, is a function of time, t in seconds.

Distance varies sinusoidally with time

• At 1 second, the distance above the ground is at its maximum of 3.7 meters from the ground.

• At 4 seconds, the distance above the ground is at its minimum, 1.5 meters from the ground

Multiple Choice

What is the average distance the head of the pump jack is from the ground (the vertical shift)?

• 1.1 m

• 3.7 m

• 2.6 m

• 1.5 m

Multiple Choice

When creating an equation to model this sinusoidal function, what would be the amplitude?

• 2.6 m

• 1.5 m

• 3.7 m

• 1.1 m

Multiple Choice

When writing an equation to model this sinusoidal function, what is the B-value?

• $$\frac{\pi}{3}$$

• $$\frac{2\pi}{3}$$

• $$6$$

• $$\frac{3\pi}{2}$$

Multiple Choice

Which would be the best graph of the sinusoidal function?

Multiple Choice

Which equation would be a model for the distance the head of the pump jack is from the ground as a function of time

• $$d=1.1\cos\left(3t-1\right)+2.6$$

• $$d=1.1\sin\left(\frac{\pi}{3}t\right)+2.6$$

• $$d=2.6\cos\frac{\pi}{3}\left(t-1\right)+1.1$$

• $$d=1.1\cos\left(\frac{\pi}{3}t-\frac{\pi}{3}\right)+2.6$$

Multiple Choice

Find the distance the head of the pump jack is from the ground at 8 seconds. Round to 2 decimals.

• 3.10 m

• 3.15 m

• 3.55 m

• 3.58 m

Multiple Choice

Find the 1st time the head of the pump jack is 3 meters above the ground. Round to 2 decimal places.

• 5.86 sec

• 0.14 sec

• 2.14 sec

• 2.25 sec

At a pier, scientists measure the depth of the water according to the tides.

The data in the table generates the given equation.

At high tide, the water depth is 8 feet.

At low tide, 6.2 hours later, the depth is 5 feet.

Multiple Choice

What is the average depth of the water?

• 6.5 feet

• 1.5 feet

• 1.57 feet

• 6.8 feet

Multiple Choice

Use the equation or table to decide: how often does the tide cycle repeat (what is the period)?

• Once every 3.1 hours.

• Once every 6.2 hours.

• Once every 9.3 hours.

• Once every 12.4 hours.

Multiple Choice

Use the equation to predict the depth after 4 hours.

• 4 feet deep

• 5.84 feet deep

• 5.42 feet deep

• 4.98 feet deep.

Multiple Choice

Use the equation to calculate how many hours it will be before the water depth is first below 7 feet?

• 9.96 hours

• 2.43 hours

• 2.91 hours

• 1.89 hours

Multiple Choice

The equation shown models the data in the table, where x stands for month of the year and y stands for daylight hours.

Use the equation to predict the length of daylight during the month of May.

• 5 hours

• 14.28 hours

• 11.9 hours

• 13.85 hours