Aim for FivePrecalculus

Modeling a periodic context

Tides at a harbor

  • 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

At a harbor, the water level is measured relative to a fixed marker on a pier. A positive water level means the water is above the marker, and a negative water level means it is below the marker.

On a certain day, the water level is highest, 7 feet above the marker, at time t = 3 hours after midnight. The water level is next at its lowest, 1 foot below the marker, at t = 9 hours. The water level continues to rise and fall in this same pattern.

The sinusoidal function h models the water level relative to the marker, in feet, as a function of time t, in hours after midnight.

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 a minimum; G is the next point where the graph crosses its midline while rising; J is the next maximum; K is the next point where the graph crosses its midline while falling; P is the next minimum.

Source: Description of a hypothetical graph

Suggested time: 18 minutes

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Part (a)

2 points

Determine possible coordinates (t, h(t)) for the five points F, G, J, K and P.

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Part (b)

2 points

The 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 points

Refer to the graph of h described above. The t-coordinate of G is t₁, and the t-coordinate of J 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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