Question

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)

Answer 1

The graph hits $\fbox{\begin\\\ \dfrac{10}{T}+2\\\end{minispace}}$ times over 2 feet for $a>2$.

Further explanation:  

The height of the wave is given by the equation as follows:  

$y=asin\left(\dfrac{2\pi t}{T}\right)$                          ......(1)

Here, $a$ is amplitude, $T$ is period of wave in second and $t$ time in seconds.  

The height $y$ of the wave is given as 2 feet and time $t$ is given as 5 seconds.  

Substitute 2 for $y$ and 5 for $t$ in equation (1).  

$2=asin\left(\dfrac{2\pi \times5}{T}\right)\\2=asin\left(\dfrac{10\pi}{T}\right)\\\dfrac{2}{a}=sin\left(\dfrac{10\pi}{T}\right)$

The above eqution is valid only for $a\geq 2$ because the maximum value of the term $sin(10\pi /T)$ is 1.  

If $T$ is the time period then in $T$ seconds the graph will hit at least 2 times over 2 feet for $a>2$.

T seconds$\rightarrow$2 hits

1 seconds $\rightarrow$ $\dfrac{2}{T}$ hits

5 seconds $\rightarrow\dfrac{2\times5}{T}$

5 seconds $\rightarrow$ $\dfrac{10}{T}$

If $T$ is time period in 5 seconds then the graph will hit $[10/T]$ times in interval 0 to $2\pi$.

Thus, the graph hits $\fbox{\begin\\\ \dfrac{10}{T}+2\\\end{minispace}}$ times over 2 feet for $a>2$.

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)  

brainly.com/question/1332667  

2. Which is the graph of f(x) = (x – 1)(x + 4)?  

brainly.com/question/2334270

 

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.