For a short time after a wave is created by wind, the height of the wave can be modeled using y = a sin 2πt/T, where a is the amplitude and T is the period of the wave in seconds. How many times over the first 5 seconds does the graph predict the wave to be 2 feet high?(SHOW WORK)

Accepted Solution

The graph hits [tex]\fbox{\begin\\\ \dfrac{10}{T}+2\\\end{minispace}}[/tex] times over 2 feet for [tex]a>2[/tex].Further explanation:  The height of the wave is given by the equation as follows:  [tex]y=asin\left(\dfrac{2\pi t}{T}\right)[/tex]                          ......(1)Here, [tex]a[/tex] is amplitude, [tex]T[/tex] is period of wave in second and [tex]t[/tex] time in seconds.  The height [tex]y[/tex] of the wave is given as 2 feet and time [tex]t[/tex] is given as 5 seconds.  Substitute 2 for [tex]y[/tex] and 5 for [tex]t[/tex] in equation (1).  [tex]2=asin\left(\dfrac{2\pi \times5}{T}\right)\\2=asin\left(\dfrac{10\pi}{T}\right)\\\dfrac{2}{a}=sin\left(\dfrac{10\pi}{T}\right)[/tex]The above eqution is valid only for [tex]a\geq 2[/tex] because the maximum value of the term [tex]sin(10\pi /T)[/tex] is 1.  If [tex]T[/tex] is the time period then in [tex]T[/tex] seconds the graph will hit at least 2 times over 2 feet for [tex]a>2[/tex].T seconds[tex]\rightarrow[/tex]2 hits1 seconds [tex]\rightarrow[/tex] [tex]\dfrac{2}{T}[/tex] hits5 seconds [tex]\rightarrow\dfrac{2\times5}{T}[/tex]5 seconds [tex]\rightarrow[/tex] [tex]\dfrac{10}{T}[/tex]If [tex]T[/tex] is time period in 5 seconds then the graph will hit [tex][10/T][/tex] times in interval 0 to [tex]2\pi[/tex].Thus, the graph hits [tex]\fbox{\begin\\\ \dfrac{10}{T}+2\\\end{minispace}}[/tex] times over 2 feet for [tex]a>2[/tex].Learn more:  1. What is the y-intercept of the quadratic function f(x) = (x – 6)(x – 2)? (0,–6) (0,12) (–8,0) (2,0)    2. Which is the graph of f(x) = (x – 1)(x + 4)?    Answer details:  Grade: High school.  Subjects: Mathematics.  Chapter: function.  Keywords: Function, wave equation, height, amplitude, equation, period, periodic function, y=asin(2pit/T), frequency, magnitude, feet, height, time period, seconds, inequality, maximum value, range, harmonic motion, oscillation, springs, strings, sonometer.