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.
$$
D(t)=
$$
$\square$

Question

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.
$$
D(t)=
$$
$\square$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Given the initial condition at $t=0$, Vlad is 1 meter behind the center, we can determine the vertical shift $d$. ‖ ‖ step 2 ⋮ Since the farthest distance behind the center is 1 meter, $d = -1$. ‖ ‖ step 3 ⋮ The swing reaches the center $\frac{\pi}{6}$ seconds later, which corresponds to the cosine function being zero. This allows us to find the horizontal stretch $b$. ‖ ‖ step 4 ⋮ The cosine function is zero at $\frac{\pi}{2}$, so $b \cdot \frac{\pi}{6} = \frac{\pi}{2}$. Solving for $b$ gives $b = 3$. ‖ ‖ step 5 ⋮ There is no information given about a vertical stretch or compression, so we assume the amplitude $a$ is 1. ‖ ‖ step 6 ⋮ The function $D(t)$ is therefore $D(t) = a \cdot \cos(b \cdot t) + d = \cos(3t) - 1$. ‖ ∻Answer∻ ⚹ $D(t) = \cos(3t) - 1$ ⚹ ∻Key Concept∻ ⚹Modeling motion with sinusoidal functions⚹ ∻Explanation∻ ⚹The horizontal distance of a swing can be modeled by a sinusoidal function, where the amplitude represents the maximum distance, the horizontal stretch/compression factor adjusts the period of the swing, and the vertical shift represents the initial position relative to the center.⚹◊What is the period of the sinusoidal function used to model the horizontal distance of the person on the swing set over time?⍭ Generate me a similar question◊

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