A buoy floating in the ocean is bobbing in simple harmonic motion with period 8 seconds and amplitude 4ft. Its displacement d from sea level at time t=0 seconds is 0ft and initially it moves downward. (Note that downward is the negative direction.) Give the equation modeling the displacement d as a function of time t. d=◻

Define Cosine Function: Since the buoy starts at 0ft and moves downward, we need a cosine function with a phase shift to represent the initial condition. The general form of the cosine function is d(t) = A * cos(B(t - C)) + D.Calculate Amplitude and Period: The amplitude A is 4ft because that's the maximum displacement from the equilibrium position.Determine Angular Frequency: The period T of the cosine function is 8 seconds. The value of B, which is the angular frequency, is related to the period by the formula B = 2π/T.Calculate B Value: Calculate B using the period T = 8 seconds: B = 2π/8 = π/4.Adjust for Initial Condition: Since the buoy starts at 0ft and moves downward, we need a negative cosine function. This means we will use -cos instead of cos in our equation.Final Displacement Equation: There is no horizontal shift (C) because the buoy starts at t=0 seconds, and there is no vertical shift (D) because the equilibrium position is at 0ft.Putting it all together, the equation for the displacement d as a function of time t is d(t) = -4 * cos(π/4 * t).

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A buoy floating in the ocean is bobbing in simple harmonic motion with period $8$ seconds and amplitude $4\,\text{ft}$ . Its displacement $d$ from sea level at time $t=0$ seconds is $0\,\text{ft}$ and initially it moves downward. (Note that downward is the negative direction.) $\newline$ Give the equation modeling the displacement $d$ as a function of time $t$ . $\newline$ $d=$ $\square$

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Grade 7

Evaluate two-variable equations: word problems

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Q. A buoy floating in the ocean is bobbing in simple harmonic motion with period

8

seconds and amplitude

4\,\text{ft}

. Its displacement

d

from sea level at time

t=0

seconds is

0\,\text{ft}

and initially it moves downward. (Note that downward is the negative direction.)

\newline

Give the equation modeling the displacement

d

as a function of time

t

\newline

d=

\square

Define Cosine Function: Since the buoy starts at $0$ ft and moves downward, we need a cosine function with a phase shift to represent the initial condition. The general form of the cosine function is $d(t) = A \cdot \cos(B(t - C)) + D$ .
Calculate Amplitude and Period: The amplitude $A$ is $4$ ft because that's the maximum displacement from the equilibrium position.
Determine Angular Frequency: The period $T$ of the cosine function is $8$ seconds. The value of $B$ , which is the angular frequency, is related to the period by the formula $B = \frac{2\pi}{T}$ .
Calculate B Value: Calculate B using the period $T = 8$ seconds: $B = 2\pi/8 = \pi/4$ .
Adjust for Initial Condition: Since the buoy starts at $0\text{ft}$ and moves downward, we need a negative cosine function. This means we will use $-\cos$ instead of $\cos$ in our equation.
Final Displacement Equation: There is no horizontal shift ( $C$ ) because the buoy starts at $t=0$ seconds, and there is no vertical shift ( $D$ ) because the equilibrium position is at $0\text{ft}$ .
Final Displacement Equation: There is no horizontal shift $C$ because the buoy starts at $t=0$ seconds, and there is no vertical shift $D$ because the equilibrium position is at $0$ ft.Putting it all together, the equation for the displacement $d$ as a function of time $t$ is $d(t) = -4 \cdot \cos(\frac{\pi}{4} \cdot t)$ .