Question

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)

Answer 1

Answer:

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

Step-by-step explanation:

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

Part A

• The amplitude is half the distance between the maximum/minimum, therefore, the amplitude is $a=\frac{4}{2}=2$

• Also, we are given that the period is 20 seconds, which means that if $\frac{2\pi}{b}=20$, then $b=\frac{\pi}{10}$

Part B

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

Part C

If you review the attached graph, you will see that when $t=15$, 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!

Answer 2

a) The amplitude of the function is 4 feet.

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

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 ($A$), in feet, is equal to the difference between highest and lowest point ($y_{max}$, $y_{min}$), in feet, divided by 2. The period ($T$), 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 ($y_{max} = 14\,ft$, $y_{min} = 6\,ft$)

$A = \frac{14\,ft-6\,ft}{2}$

$A = 4\,ft$

The amplitude of the function is 4 feet. $\blacksquare$

Period

The period of the function is 20 seconds. $\blacksquare$

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

$y(t) = y_{o} + A \cdot \sin \frac{2\pi\cdot t}{T}$ (1)

Where:

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

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

$y(t) = 10 +4\cdot \sin \frac{\pi\cdot t}{10}$ (2)

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

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. $\blacksquare$

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