a pendulum is swinging next to a wall. the distance d(t) (in cm) between the bob of the pendulum and the…

a pendulum is swinging next to a wall. the distance d(t) (in cm) between the bob of the pendulum and the wall as a function of time t (in seconds) can be modeled by a sinusoidal expression of the form a * sin(b * t)+d. at t = 0, when the pendulum is exactly in the middle of its swing, the bob is 5 cm away from the wall. the bob reaches the closest point to the wall, which is 3 cm from the wall, 1 second later. find d(t). t should be in radians. d(t)=

a pendulum is swinging next to a wall. the distance d(t) (in cm) between the bob of the pendulum and the wall as a function of time t (in seconds) can be modeled by a sinusoidal expression of the form a * sin(b * t)+d. at t = 0, when the pendulum is exactly in the middle of its swing, the bob is 5 cm away from the wall. the bob reaches the closest point to the wall, which is 3 cm from the wall, 1 second later. find d(t). t should be in radians. d(t)=

Answer

Explanation:

Step1: Determine the amplitude $a$

The amplitude is half the difference between the maximum and minimum distances. The maximum distance is $5$ cm and the minimum is $3$ cm. So $a=\frac{5 - 3}{2}=1$.

Step2: Determine the vertical - shift $k$

The vertical - shift $k$ is the average of the maximum and minimum distances. So $k=\frac{5 + 3}{2}=4$.

Step3: Determine the value of $b$

The pendulum goes from the middle of its swing to the closest point in $1$ second. A sine function goes from its mid - value to its minimum value in $\frac{3}{4}$ of a period or $\frac{1}{4}$ of a period (depending on the phase). Since it goes from the mid - value to the minimum value, it is $\frac{1}{4}$ of a period. If $\frac{T}{4}=1$, then the period $T = 4$. The formula for the period of a sine function $y = a\sin(bx)+k$ is $T=\frac{2\pi}{b}$. Substituting $T = 4$ into $T=\frac{2\pi}{b}$, we get $4=\frac{2\pi}{b}$, and solving for $b$ gives $b=\frac{\pi}{2}$.

Step4: Write the function

Since at $t = 0$, the pendulum is in the middle of its swing and we are using a sine function of the form $D(t)=a\sin(bt)+k$, substituting $a = 1$, $b=\frac{\pi}{2}$, and $k = 4$ into the formula, we get $D(t)=1\times\sin(\frac{\pi}{2}t)+4$.

Answer:

$D(t)=\sin(\frac{\pi}{2}t)+4$