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Modeling with Periodic Functions Answers

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The height, d, in feet of a ball suspended from a spring as a function of time, t, in seconds can be modeled by the equation . The ball is released from its lowest point at seconds. Using your knowledge of the general form of sine and cosine functions, which of the following equations can also model this situation?

Question illustration
A
d = negative 2 cosine (pi t) + 5
Option A
B
d = negative 2 cosine (pi (t + one-half)) + 5
Option B
C
d = 2 cosine (pi t) + 5
Option C
D
d = 2 cosine (pi (t + one-half)) + 5
Option D
2
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The displacement, d, in millimeters of a tuning fork as a function of time, t, in seconds can be modeled with the equation . What is the maximum displacement of the tuning fork?

Question illustration
A
0.2 mm
B
0.4 mm
C
0.8 mm
D
2.5 mm
3

Blood pressure can be modeled by a sinusoidal curve, where the maximum and minimum on the curve correlate to the person’s blood pressure reading. Henry’s blood pressure is modeled by the function , where t is time in seconds.Which graph accurately depicts this model?

Question illustration
A
On a coordinate plane, seconds is on the x-axis and blood pressure is on the y-axis. A curve starts at (0, 100). It has a maximum of y = 130 and a minimum of y = 70. It goes through one cycle in 1 second.
Option A
B
On a coordinate plane, seconds is on the x-axis and blood pressure is on the y-axis. A curve starts at (0, 100). It has a maximum of y = 130 and a minimum of y = 70. It goes through one cycle in 2 seconds.
Option B
C
On a coordinate plane, seconds is on the x-axis and blood pressure is on the y-axis. A curve starts at (0, 0). It has a maximum of y = 30 and a minimum of y = negative 30. It goes through one cycle in 1 second.
Option C
D
On a coordinate plane, seconds is on the x-axis and blood pressure is on the y-axis. A curve starts at (0, 30). It has a maximum of y = 60 and a minimum of y = negative 0. It goes through one cycle in 1 second.
Option D
4

The height, h, in feet of a flag on one blade of a windmill as a function of time, t, in seconds can be modeled by the equation . What is the minimum height of the flag?

Question illustration
A
3 feet
B
9 feet
C
12 feet
D
15 feet
5

The equation models the horizontal distance, d, in inches of the pendulum of a grandfather clock from the center as it swings from right to left and left to right as a function of time, t, in seconds. According to the model, how long does it take for the pendulum to swing from its rightmost position to its leftmost position and back again? Assume that right of center is a positive distance and left of center is a negative distance.

Question illustration
A
0.625 seconds
B
0.8 seconds
C
1.25 seconds
D
1.6 seconds
6

A rider boards a Ferris wheel 10 feet above ground level. The diameter of the Ferris wheel is 50 feet and the wheel completes one full rotation every 24 seconds. Which graph represents the relationship between time and height off the ground?

A
On a coordinate plane, a curve starts at (0, 10). It has a maximum at y = 60 and a minimum at y = 10. It increases through the y-axis. Its cycle is 0.4 x.
Option A
B
On a coordinate plane, a curve starts at (0, 60). It has a maximum at y = 60 and a minimum at y = 10. It decreases through the y-axis. Its cycle is 0.4 x.
Option B
C
On a coordinate plane, a curve starts at (0, 10). It has a maximum at y = 60 and a minimum at y = 10. It increases through the y-axis. Its cycle is 0.8 x.
Option C
D
On a coordinate plane, a curve starts at (0, 10). It has a maximum at y = 60 and a minimum at y = 10. It decreases through the y-axis. Its cycle is 0.8 x.
Option D
7

The distance, d, in inches of a weight attached to a spring from its equilibrium as a function of time, t, in seconds can be modeled by the graph below. Which equation is represented in the graph below?

Question illustration
A
d = negative 10 cosine (StartFraction pi Over 2 EndFraction t)
Option A
B
d = negative 10 cosine (pi t)
Option B
C
d = negative 5 cosine (StartFraction pi Over 2 EndFraction t)
Option C
D
d = negative 5 cosine (pi t)
Option D
8

The height, h, in feet of a piece of cloth tied to a waterwheel in relation to sea level as a function of time, t, in seconds can be modeled by the equation . How long does it take for the waterwheel to complete one turn?

Question illustration
A
5 seconds
B
10 seconds
C
20 seconds
D
40 seconds
9

Throughout the day, the depth of water at the end of a dock varies with the tides. The function represents the height, in feet, of the water t hours after midnight.Which graph shows the height of the water at the dock at any time after midnight?

Question illustration
A
On a coordinate plane, feet is on the x-axis and depth is on the y-axis. A curve starts at (0, 5). It decreases through the y-axis. It has a minimum of y = 2 and a maximum of y =12. It goes through 1 cycle in 13 feet.
Option A
B
On a coordinate plane, feet is on the x-axis and depth is on the y-axis. A curve starts at (0, 5). It increases through the y-axis. It has a minimum of y = 2 and a maximum of y =12. It goes through 1 cycle in 13 feet.
Option B
C
On a coordinate plane, feet is on the x-axis and depth is on the y-axis. A curve starts at (0, 5). It increases through the y-axis. It has a minimum of y = 2 and a maximum of y =12. It goes through 1 cycle in 7 feet.
Option C
D
On a coordinate plane, feet is on the x-axis and depth is on the y-axis. A curve starts at (0, negative 2). It increases through the y-axis. It has a minimum of y = negative 5 and a maximum of y =5. It goes through 1 cycle in 13 feet.
Option D
10

The equation can be used to model the height, h, in feet of the end of one blade of a windmill turning on an axis above the ground as a function of time, t, in seconds. How long is the blade? Assume that the blade is pointing to the right, parallel to the ground, at t = 0, and that the windmill turns counterclockwise at a constant rate.

Question illustration
A
7 feet
B
14 feet
C
21 feet
D
28 feet

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