Question

Math

Posted 6 months ago

```
An object is attached by a string to the end of a spring. Fang throws the object upwards and starts a stopwatch at $t=0$ seconds. The object starts oscillating vertically in a periodic way that can be modeled by a trigonometric function.
The object's average height is $-20 \mathrm{~cm}$ (measured from the top of the spring). It first achieves that average height on the way up at $t=0.2$ seconds, and then again every 2 seconds. The object's maximum and minimum heights are each $5 \mathrm{~cm}$ from its average height.
Find the formula of the trigonometric function that models the height $H$ of the weight $t$ seconds after Fang started the stopwatch. Define the function using radians.
$H(t)=$
$\square$
What is the height of the object after 0.6 seconds? Round your answer, if necessary, to two decimal places.
$\square$ $\mathrm{cm}$
```

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Answer from Sia

Posted 6 months ago

Solution by Steps

step 1

To find the trigonometric function that models the height $H$ of the weight over time, we need to determine the amplitude, period, phase shift, and vertical shift

step 2

The amplitude is the maximum deviation from the average height, which is $5 \mathrm{~cm}$

step 3

The period is the time it takes for one complete oscillation, which is $2$ seconds. Since the function will be in terms of radians, we use the relationship $T = \frac{2\pi}{\omega}$ to find the angular frequency $\omega$

step 4

Solving for $\omega$, we get $\omega = \frac{2\pi}{T} = \frac{2\pi}{2} = \pi$

step 5

The phase shift is determined by the time it first reaches the average height on the way up. Since this occurs at $t = 0.2$ seconds, the phase shift is $0.2$ seconds to the right. In terms of $\omega$, the phase shift is $\frac{\omega}{5} = \frac{\pi}{5}$

step 6

The vertical shift is the average height, which is $-20 \mathrm{~cm}$

step 7

Combining these values, the trigonometric function is $H(t) = 5 \cos(\pi t - \frac{\pi}{5}) - 20$

step 8

To find the height of the object after $0.6$ seconds, we substitute $t = 0.6$ into the function

step 9

Calculating $H(0.6) = 5 \cos(\pi \cdot 0.6 - \frac{\pi}{5}) - 20$

step 10

Simplifying the cosine term, we get $H(0.6) = 5 \cos(\frac{3\pi}{5} - \frac{\pi}{5}) - 20 = 5 \cos(\frac{2\pi}{5}) - 20$

step 11

Using a calculator, we find that $\cos(\frac{2\pi}{5}) \approx 0.309$

step 12

Therefore, $H(0.6) \approx 5 \cdot 0.309 - 20 \approx -18.55 \mathrm{~cm}$

[1] Answer

$H(t) = 5 \cos(\pi t - \frac{\pi}{5}) - 20$

$H(0.6) \approx -18.55 \mathrm{~cm}$

Key Concept

Modeling periodic motion with trigonometric functions

Explanation

The trigonometric function for the height of the weight over time is determined by the amplitude, period, phase shift, and vertical shift. The height at a specific time is found by substituting the time into the function.

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