Modeling a periodic context
A buoy on the waves
- Unit 3
- 6 points
- About 18 minutes
A repeating real-world situation is modeled by a sinusoidal function. You find coordinates of key points on its graph, write the function's equation, and describe how the function, its concavity and its rate of change behave on an interval. No calculator. On the exam: Question 3 of 4. Part B of the free-response section: 2 questions in 35 minutes, no calculator allowed.
The question and its sources
No calculator is allowed for this question. Angle measures are in radians. Give exact values (fractions and multiples of π are fine). On the exam this question shows a graph; here the graph is described in words.
The situation
A buoy floats on the surface of a lake as waves pass by, so it moves up and down. Its highest position is 0.6 meter above its rest position, and its lowest position is 0.6 meter below its rest position. The buoy completes one full up-and-down cycle every 5 seconds.
At time t = 0 seconds, the buoy is at its rest position and moving downward.
The sinusoidal function h models the buoy's vertical position relative to its rest position, in meters, as a function of time t, in seconds. A positive value of h(t) means the buoy is above its rest position; a negative value means it is below.
Source: Hypothetical scenario
Graph of h (described)
The graph of h and its dashed midline are drawn for two full cycles. No axes or scale are shown.
Five points, F, G, J, K and P, are labeled on the graph in order from left to right, one after another: F is where the graph crosses its midline while falling; G is the next minimum; J is the next point where the graph crosses its midline while rising; K is the next maximum; P is the next point where the graph crosses its midline while falling.
Source: Description of a hypothetical graph
Suggested time: 18 minutes
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Part (a)
2 pointsDetermine possible coordinates (t, h(t)) for the five points F, G, J, K and P.
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Part (b)
2 pointsThe function h can be written in the form h(t) = a sin(b(t + c)) + d. Find values of the constants a, b, c and d.
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Part (c)
2 pointsRefer to the graph of h described above. The t-coordinate of F is t₁, and the t-coordinate of G is t₂. (i) On the interval (t₁, t₂), which of the following is true about h? (a) h is positive and increasing. (b) h is positive and decreasing. (c) h is negative and increasing. (d) h is negative and decreasing. (ii) On the interval (t₁, t₂), describe the concavity of the graph of h and determine whether the rate of change of h is increasing or decreasing.
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