Question

Math

Posted 4 months ago

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Vlad is playing on a swing set.
His horizontal distance $D(t)$ (in $\mathrm{m}$ ) from the center (where being behind the center means a negative distance) as a function of time $t$ (in seconds) can be modeled by a sinusoidal expression of the form $a \cdot \cos (b \cdot t)+d$.
At $t=0$, when he pushes off, he is $1 \mathrm{~m}$ behind the center, which is as far back as he goes. The swing reaches the center $\frac{\pi}{6}$ seconds later.
Find $D(t)$.
$t$ should be in radians.
$D(t)=$
$\square$
```

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Answer from Sia

Posted 4 months ago

Solution by Steps

step 1

To find the function $D(t)$, we need to determine the values of $a$, $b$, and $d$ in the sinusoidal function $a \cdot \cos(b \cdot t) + d$

step 2

Given that at $t = 0$, Vlad is 1 meter behind the center, we can determine $d$ because this is the farthest point back he goes, which is the amplitude of the cosine function. Thus, $d = -1$

step 3

Since the swing reaches the center $\frac{\pi}{6}$ seconds later, we can find $b$ by using the fact that the cosine function completes a half-cycle when the swing goes from the farthest point back to the center. Therefore, $b \cdot \frac{\pi}{6} = \frac{\pi}{2}$, which gives us $b = 3$

step 4

The amplitude $a$ is the maximum distance from the center, which is also 1 meter in this case, but since we already accounted for the negative sign in $d$, we have $a = 1$

step 5

Combining the values of $a$, $b$, and $d$, we get the function $D(t) = 1 \cdot \cos(3 \cdot t) - 1$

Answer

$D(t) = \cos(3t) - 1$

Key Concept

Determining parameters of a sinusoidal function

Explanation

The amplitude, period, and phase shift of a sinusoidal function can be determined from given points on the function's graph. In this case, the amplitude is the maximum distance from the center, the period is related to the time it takes to reach the center from the farthest point, and the phase shift is zero since the function starts at the maximum point.

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