Valeria is playing with her accordion. The length of the accordion A(t) (in cm ) after she starts playing as a function of time t (in seconds) can be modeled by a sinusoidal expression of the form a*cos(b*t)+d At t=0, when she starts playing, the accordion is 15cm long, which is the shortest it gets. 1.5 seconds later the accordion is at its average length of 21cm. Find A(t). t should be in radians. A(t)=◻

Determine Amplitude: Determine the amplitude (a) of the sinusoidal function. Since the accordion starts at its shortest length of 15 cm and goes to its average length of 21 cm, the amplitude is the difference between the average length and the shortest length. a = 21 cm - 15 cm = 6 cmDetermine Vertical Shift: Determine the vertical shift (d) of the sinusoidal function. The vertical shift is the average length of the accordion, which is given as 21 cm. d = 21 cmDetermine Angular Frequency: Determine the angular frequency (b) of the sinusoidal function. We know that the accordion reaches its average length after 1.5 seconds, which corresponds to half of the period of the sinusoidal function. Therefore, the period (T) is 2 * 1.5 seconds = 3 seconds. The angular frequency b is related to the period by the formula b = 2π/T. b = 2π / 3Write Sinusoidal Function: Write the sinusoidal function A(t). Since the accordion starts at its shortest length, we know that the cosine function starts at its minimum value. Therefore, we need to reflect the cosine function vertically to match the behavior of the accordion's length. This means we will use a negative amplitude in our function. A(t) = -a * cos(b * t) + d Substitute the values of a, b, and d into the equation. A(t) = -6 * cos((2π/3) * t) + 21

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Valeria is playing with her accordion.
The length of the accordion
A(t) (in
cm ) after she starts playing as a function of time
t (in seconds) can be modeled by a sinusoidal expression of the form
a*cos(b*t)+d
At
t=0, when she starts playing, the accordion is
15cm long, which is the shortest it gets. 1.5 seconds later the accordion is at its average length of
21cm.
Find
A(t).

t should be in radians.

A(t)=◻

Valeria is playing with her accordion. $\newline$ The length of the accordion $A(t)$ (in $\mathrm{cm}$ ) after she starts playing as a function of time $t$ (in seconds) can be modeled by a sinusoidal expression of the form $a \cdot \cos (b \cdot t)+d$ . $\newline$ At $t=0$ , when she starts playing, the accordion is $15 \mathrm{~cm}$ long, which is the shortest it gets. $1$ . $5$ seconds later the accordion is at its average length of $21 \mathrm{~cm}$ . $\newline$ Find $A(t)$ . $\newline$ $t$ should be in radians. $\newline$ $A(t)=\square$

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Q. Valeria is playing with her accordion.

\newline

The length of the accordion

A(t)

(in

\mathrm{cm}

) after she starts playing 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

\newline

t=0

, when she starts playing, the accordion is

15 \mathrm{~cm}

long, which is the shortest it gets.

1

5

seconds later the accordion is at its average length of

21 \mathrm{~cm}

\newline

Find

A(t)

\newline

t

should be in radians.

\newline

A(t)=\square

Determine Amplitude: Determine the amplitude $a$ of the sinusoidal function. $\newline$ Since the accordion starts at its shortest length of $15\,\text{cm}$ and goes to its average length of $21\,\text{cm}$ , the amplitude is the difference between the average length and the shortest length. $\newline$ $a = 21\,\text{cm} - 15\,\text{cm} = 6\,\text{cm}$
Determine Vertical Shift: Determine the vertical shift $d$ of the sinusoidal function. $\newline$ The vertical shift is the average length of the accordion, which is given as $21$ cm. $\newline$ $d = 21$ cm
Determine Angular Frequency: Determine the angular frequency $b$ of the sinusoidal function. $\newline$ We know that the accordion reaches its average length after $1.5$ seconds, which corresponds to half of the period of the sinusoidal function. Therefore, the period $T$ is $2 \times 1.5$ seconds = $3$ seconds. $\newline$ The angular frequency $b$ is related to the period by the formula $b = \frac{2\pi}{T}$ . $\newline$ $b = \frac{2\pi}{3}$
Write Sinusoidal Function: Write the sinusoidal function $A(t)$ . Since the accordion starts at its shortest length, we know that the cosine function starts at its minimum value. Therefore, we need to reflect the cosine function vertically to match the behavior of the accordion's length. This means we will use a negative amplitude in our function. $A(t) = -a \cdot \cos(b \cdot t) + d$ Substitute the values of $a$ , $b$ , and $d$ into the equation. $A(t) = -6 \cdot \cos\left(\frac{2\pi}{3} \cdot t\right) + 21$