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A message in a bottle is floating on top of the ocean in a periodic manner. The time between periods of maximum heights is 20 seconds, and the average height of the bottle is 10 feet. The bottle moves in a manner such that the distance from the highest and lowest point is 4 feet. A cosine function can model the movement of the message in a bottle in relation to the height.

Part A: Determine the amplitude and period of the function that could model the height of the message in a bottle as a function of time, t. (5 points)

Part B: Assuming that at t = 0 the message in a bottle is at its average height and moves upwards after, what is the equation of the function that could represent the situation? (5 points)

Part C: Based on the graph of the function, after how many seconds will it reach its lowest height? (5 points)

Sagot :

Answer:

See below for answers and explanations for each part (along with attached graph)

Step-by-step explanation:

Recall the equation [tex]f(x)=asin(bx+c)+d[/tex] where [tex]a[/tex] is the amplitude, [tex]\frac{2\pi}{b}[/tex] is the period, [tex]-\frac{c}{b}[/tex] is the phase/vertical shift, and [tex]d[/tex] is the midline

Part A

  • The amplitude is half the distance between the maximum/minimum, therefore, the amplitude is [tex]a=\frac{4}{2}=2[/tex]
  • Also, we are given that the period is 20 seconds, which means that if [tex]\frac{2\pi}{b}=20[/tex], then [tex]b=\frac{\pi}{10}[/tex]

Part B

  • Since [tex]f(x)=sin(x)[/tex] begins at its average and we use the identity [tex]sin(x)=cos(x-\frac{\pi}{2})[/tex] to represent the wave traveling up first, then we have [tex]c=-\frac{\pi}{2}[/tex], making the phase shift [tex]-\frac{c}{b}=-\frac{-\frac{\pi}{2}}{\frac{\pi}{10}}=5[/tex], or 5 feet to the right
  • We also know that the average, or the midline, must be [tex]d=10[/tex], making our equation [tex]f(t)=2cos(\frac{\pi}{10}t-\frac{\pi}{2} )+10[/tex] as a function of [tex]t[/tex]

Part C

If you review the attached graph, you will see that when [tex]t=15[/tex], then the bottle will reach its lowest height of 8 feet

I hope these explanations and the attached graph help you understand sinusoidal functions better! Please mark this answer brainliest if you found this answer helpful!

View image goddessboi
xero099

a) The amplitude of the function is 4 feet.

b) The function that represents the situation is [tex]y(t) = 10 +4\cdot \sin \frac{\pi\cdot t}{10}[/tex].

c) The bottle will take 15 seconds to reach its lowest height.

How to find a function for the height of a bottle and how to analyze its motion

a) The amplitude ([tex]A[/tex]), in feet, is equal to the difference between highest and lowest point ([tex]y_{max}[/tex], [tex]y_{min}[/tex]), in feet, divided by 2. The period ([tex]T[/tex]), in seconds, is the time taken by the bottle to complete one cycle. In this case, the period is the time between two maxima. Hence, we proceed to determine each variable:

Amplitude ([tex]y_{max} = 14\,ft[/tex], [tex]y_{min} = 6\,ft[/tex])

[tex]A = \frac{14\,ft-6\,ft}{2}[/tex]

[tex]A = 4\,ft[/tex]

The amplitude of the function is 4 feet. [tex]\blacksquare[/tex]

Period

The period of the function is 20 seconds. [tex]\blacksquare[/tex]

b) The function that represents the situation is based on this model:

[tex]y(t) = y_{o} + A \cdot \sin \frac{2\pi\cdot t}{T}[/tex] (1)

Where:

  • [tex]y_{o}[/tex] - Average height of the bottle, in feet.
  • [tex]t[/tex] - Time, in seconds.
  • [tex]y(t)[/tex] - Current height, in feet.

If we know that [tex]A = 4\,ft[/tex], [tex]y_{o} = 10\,ft[/tex] and [tex]T = 20\,s[/tex], then the function that represents the situation is:

[tex]y(t) = 10 +4\cdot \sin \frac{\pi\cdot t}{10}[/tex] (2)

The function that represents the situation is [tex]y(t) = 10 +4\cdot \sin \frac{\pi\cdot t}{10}[/tex]. [tex]\blacksquare[/tex]

c) With the help of a graphic tool, we graph the function in time. According to the image, the bottle will take 15 seconds to reach its lowest height. [tex]\blacksquare[/tex]

To learn more on simple harmonic motion, we kindly invite to check this verified question: https://brainly.com/question/17315536  

View image xero099
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