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3 years ago

Find the value of each variable

Mathematics

1 answer:

UNO [17]3 years ago

7 0

I'm assuming that they are perpendicular, but to be safe I'll say they are supplementary.

180 = 7x - 248 + x + 44
180 = 8x - 204
384 = 8x
/8 /8
48 = x

180 = 2y + 50 + 5y - 17
180 = 7y + 33
147 = 7y
/7 /7
21 = y

Therefore x = 48 and y = 21. If I made any errors please say so and don't report me :3

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1..A message in a bottle is floating on top of the ocean in a periodic manner. The time between periods of maximum heights is 26

statuscvo [17]

Answer:

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

Step-by-step explanation:

Part A

The amplitude of a sinusoidal function is half the distance between the maximum and the minimum. It is given to us that the distance from the highest and lowest point is 6 feet, so our amplitude is 6/2 = 3 feet

Part B

The graph's function would be in the form of $y=acos(bx+c)+d$ where $a$ is the amplitude, $\frac{2\pi}{b}$ is the period, $-\frac{c}{b}$ is the phase/horizontal shift, and $d$ is the average/midline.

We already know our amplitude of $a=3$ from part A.

Since our period is given to us as 26 seconds, then we can use the equation $\frac{2\pi}{b}=26$ to find $b$ , which happens to be $b=\frac{\pi}{13}$ .

Since the cosine function starts at its maximum and we want it to start at the average where the bottle travels up, we would need to use the cofunction identity $sin(x)=cos(x-\frac{\pi}{2})$ which shifts the cosine graph $\frac{\pi}{2}$ units to the right. This means that $c=-\frac{\pi}{2}$ , making our phase shift $-\frac{c}{b}=-\frac{-\frac{\pi}{2}}{\frac{\pi}{13}}=6.5$ , or 6.5 feet to the right

Our average/midline would be $d=12$ as given as the average height by the problem.

Therefore, the function is $f(x)=3cos(\frac{\pi}{13}x-\frac{\pi}{2})+12$

Part C

Using our determined function from Part B, by looking at its graph, we see that the bottle will reach its lowest height of 9 feet after 19.5 seconds (see attached graph).