valeria is playing with her accordion. the length of the accordion a(t) (in cm) after she starts playing as…

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 15 cm long, which is the shortest it gets. 1.5 seconds later the accordion is at its average length of 21 cm. find a(t). t should be in radians. a(t) =

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 15 cm long, which is the shortest it gets. 1.5 seconds later the accordion is at its average length of 21 cm. find a(t). t should be in radians. a(t) =

Answer

Answer:

$A(t)=6\cos\left(\frac{\pi}{3}t\right)+21$

Explanation:

Step1: Find the value of $d$

The average value of a cosine - type function $y = a\cos(bt)+d$ is $d$. Given that the average length of the accordion is $21$ cm, so $d = 21$.

Step2: Find the value of $a$

The amplitude $a$ is the distance from the average value to the maximum or minimum value. The minimum value of the accordion length is $15$ cm and the average value is $21$ cm. So $|a|=21 - 15=6$. Since at $t = 0$ the function reaches its minimum, $a=- 6$.

Step3: Find the value of $b$

We know that at $t = 0$, $A(0)=15$ and at $t = 1.5$, $A(1.5)=21$. The general form of the cosine function is $A(t)=a\cos(bt)+d$. Substituting $a=-6$, $d = 21$ into $A(t)$, we get $A(t)=-6\cos(bt)+21$. When $t = 1.5$, $A(1.5)=-6\cos(1.5b)+21 = 21$. Then $\cos(1.5b)=0$. The cosine function $\cos(x)=0$ when $x=\frac{\pi}{2}+k\pi,k\in\mathbb{Z}$. Let's take the simplest non - zero case, when $1.5b=\frac{\pi}{2}$, then $b=\frac{\pi}{3}$. So the function $A(t)=6\cos\left(\frac{\pi}{3}t\right)+21$.