Module 5: Lesson 2

 

Self-Check 1

 

Questions 2, 6.a., 6.d., 13, and 15 on pages 275 to 278 of the textbook

  1.  
    1. x ≈ 1.3, 4.5
    2. x ≈ −3, −1.9, 0.1, 1.2, 3.2, 4.1, 6.3, 7.2
  1.  
    1. Domain: {t|t ≥ 0, t ∈ R}
      Range: {P|2000 ≤ P ≤ 14 000, P ∈ N} (Note that the population needs to be in the Natural Number System because you cannot have decimal values for people.)
    1. Domain: {t|t ≥ 0, t ∈ R}
      Range: {h|5 ≤ h ≤ 23, h ∈ R}
  1.  
    1. This answer can be solved both algebraically and graphically, although at this point it is likely that students will solve the question graphically. A possible sketch of the graph is shown, using a domain of {t|0 ≤ t ≤ 24, t ∈ R}.

       
      Two graphs are shown in the diagram. There is a sinusoidal graph with the equation F(t) equals 500 times the sine of pi times x divided by 12 all plus 1000. There is also a horizontal linear equation through y = 650. The two graphs intersect in more than one place, but the first intersection point occurs at (14.96, 650).

      The population will drop below 650 after about 15 months.
    2. Two graphs are shown in the diagram. There is a sinusoidal graph with the equation F(t) equals 500 times the sine of pi times x divided by 12 all plus 1000. There is also a sinusoidal graph L(t) equals 5000 times the sine of pi divided by 12 times t minus 12 all plus 10 000. 

      The maximum fox population is 1500, occurring in the 6th month; the minimum fox population is 500, occurring in the 18th month. The maximum population of lemmings is 15 000, occurring in the 18th month; the minimum population of lemmings is 5000, occurring in the 6th month.
    3. The maximum population for the foxes occurs when the population for the lemmings is at a minimum, and vice versa. This appears to follow a typical predator-prey cycle found in many instances in the wild. The population of the predator is directly dependent upon the availability of the prey; as a result, it makes sense that the two populations are directly tied to each other.
  1.  
    1. The maximum height above the horizon is 7.5 Sun widths; the minimum height above the horizon is 1 Sun width.
    2. The period is 24 h.
    3. Because the period is 24 h, you know that the b value is

      The amplitude is This means that the amplitude is

      The graph appears to be a sinusoidal graph that has been reflected about the x-axis; this means that you can state that the a value is −3.25. If this reflection is used, there will be no phase shift.

      The vertical displacement is or

      With this knowledge, you can state that a sinusoidal equation that models the midnight Sun is where y represents the number of Sun widths and x represents the time in hours.


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