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A pendulum is swinging next to a wall. The distance from the bob of the swinging pendulum to the wall varies in a periodic way that can be modeled by a trigonometric function.

The function has a period of 0.8 seconds, amplitude of 6 cm, and midline [tex]$H=15$[/tex] cm. At time [tex]$t=0.5$[/tex] seconds, the bob is at its midline, moving towards the wall.

Find the formula of the trigonometric function that models the distance [tex]$H$[/tex] from the pendulum's bob to the wall after [tex]$t$[/tex] seconds. Define the function using radians.

[tex]\[ H(t) = \square \][/tex]

Sagot :

GinnyAnsweridn
To find the trigonometric function that models the distance [tex]\( H \)[/tex] from the pendulum's bob to the wall after [tex]\( t \)[/tex] seconds, given the information:

- Period [tex]\( T = 0.8 \)[/tex] seconds
- Amplitude [tex]\( A = 6 \)[/tex] cm
- Midline [tex]\( H = 15 \)[/tex] cm
- At [tex]\( t = 0.5 \)[/tex] seconds, the bob is at its midline and moving towards the wall.

We can proceed as follows:

### Step 1: Determine the Angular Frequency
The angular frequency [tex]\( \omega \)[/tex] is calculated using the formula:
[tex]\[ \omega = \frac{2\pi}{T} \][/tex]
Given the period [tex]\( T = 0.8 \)[/tex] seconds:
[tex]\[ \omega = \frac{2\pi}{0.8} = 7.853981633974483 \][/tex]

### Step 2: Formulate the Function Based on Trigonometric Characteristics
The displacement of the pendulum from the midline can be represented by a cosine function because at [tex]\( t = 0.5 \)[/tex] seconds, the bob is at the midline and moving towards the wall, which aligns well with the behavior of the cosine function offset by [tex]\(\pi/2\)[/tex] phase shift:
[tex]\[ H(t) = A \cos(\omega t + \phi) + \text{midline} \][/tex]

### Step 3: Determine the Phase Shift [tex]\(\phi\)[/tex]
Since the bob is at the midline at [tex]\( t = 0.5 \)[/tex] seconds and moving towards the wall, let's set:
[tex]\[ \cos(\omega \cdot t + \phi) = 0 \][/tex]
At [tex]\( t = 0.5 \)[/tex] seconds:
[tex]\[ \cos(7.853981633974483 \cdot 0.5 + \phi) = 0 \][/tex]
Solving for [tex]\(\phi\)[/tex]:
[tex]\[ 7.853981633974483 \cdot 0.5 + \phi = \frac{\pi}{2} \][/tex]
[tex]\[ \phi = \frac{\pi}{2} - 3.9269908169872415 = -1.5707963267948966 \][/tex]

### Step 4: Write the Final Equation
Now we combine all the values into the cosine function to represent the distance [tex]\( H(t) \)[/tex].

Thus, the distance [tex]\( H(t) \)[/tex] from the pendulum's bob to the wall is:
[tex]\[ H(t) = 6 \cos(7.853981633974483 \cdot t - 1.5707963267948966) + 15 \][/tex]

Therefore, the formula for [tex]\( H(t) \)[/tex] is:
[tex]\[ H(t) = 6 \cos(7.853981633974483 t - 1.5707963267948966) + 15 \][/tex]
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