Question

Math

Posted 3 months ago

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

Sign in to unlock the answer

Answer from Sia

Posted 3 months ago

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.

Not the question you are looking for? Ask here!

Enter question by text

Enter question by image

Upgrade to Asksia Pro

Join a AskSia's Pro Plan, and get 24/7 AI tutoring for your reviews, assignments, quizzes and exam preps.

Unlimited chat query usages

Strong algorithms that better know you

Early access to new release features

Study Other Question