A weight suspended by a spring vibrates vertically according to the function D D given by D(t)=2sin(4π(t+18)) D ( t ) = 2 sin ( 4 π ( t + 1 8 ) ) , where D(t) D ( t ) , in centimeters, is the directed distance of the weight from its central position t t seconds after the start of the motion. Assume the positive direction is upward. What is the instantaneous rate of change of the weight’s position, in centimeters per second, at the moment the weight is first 1 centimeter above its central position?

[tex]D(t)=2Sin (4\pi(t+18))\\1=2Sin (4\pi(t+18))\\\frac{1}{2}=Sin(4\pi t +72\pi)\\4\pi t + 72 \pi = \frac{\pi}{6}\\4\pi t = \frac{\pi}{6}-72\pi\\t=\frac{\frac{\pi}{6}-72\pi}{4\pi}[/tex]

This is the time at which this occurs.

To find instantaneous rate of change, we differentiate D(t) and plug in this t we found. Remembering that d/dt (Sin t) = Cos t

[tex]D(t)=2Sin (4\pi(t+18))\\D(t)=2Sin(4\pi t + 72\pi)\\D'(t)=2Cos(4\pi t + 72\pi)(4\pi)[/tex]

Now putting t:

[tex]D'(t)=2Cos(4\pi t + 72\pi)(4\pi)\\D'(\frac{\frac{\pi}{6}-72\pi}{4\pi})=2Cos(4\pi (\frac{\frac{\pi}{6}-72\pi}{4\pi}) + 72\pi)(4\pi)\\=8\pi Cos(\frac{\pi}{6})\\=21.75[/tex]

Thus, the instantaneous rate would be around 21.75 cm/s

ahirohit963 ahirohit963 · Answer 2 · 2021-10-22T15:12:33+02:00

The instantaneous rate is 21.75 cm/sec.

Step-by-step explanation:

Given :

[tex]\rm D(t) = 2sin(4\pi(t+18))[/tex] --- (1)

Solution :

Given that the weight is first 1 centimeter above its central position, that means D(t) = 1. From equation (1) we get

[tex]\rm 1 = 2sin(4\pi(t+18))[/tex]

[tex]\rm 0.5 =sin(4\pi(t+18))[/tex]

[tex]\rm \dfrac{\pi }{6} =(4\pi(t+18))[/tex]

[tex]\rm t = \dfrac{\dfrac{\pi}{6}-72\pi}{4\pi}[/tex]

This is the time at which this occur.

Now differentiate equation (1) with respect to time t,

[tex]\rm D'(t) = 2 cos(4\pi(t+18))(4\pi)[/tex] ---- (2)

Now put the value of t in equation (2),

[tex]\rm D'(\dfrac{\dfrac{\pi}{6}-72\pi}{4\pi})=2cos(4\pi(\dfrac{\dfrac{\pi}{6}-72\pi}{4\pi}+18))(4\pi)[/tex]

[tex]\rm= 8\pi cos(\dfrac{\pi}{6})[/tex]

= 21.75 cm/sec

Therefore, the instantaneous rate is 21.75 cm/sec.

For more information, refer the link given below

https://brainly.com/question/19573890?referrer=searchResults