Question

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Answer 1

Step-by-step explanation

From the statement, we know that the height h (in feet) of an object is given by the following function of time t (in seconds):

`h(t)=3.0+2.7\cdot\sin(1.3t+0.9).`

a. The velocity of the object is obtained by taking the first derivative:

`\begin{gathered} v(t)=h^(\prime)(t)=(d)/(dt)(3.0+2.7\cdot\sin(1.3t+0.9)) \\ =0+2.7\cdot(d)/(dt)\sin(1.3t+0.9)), \\ =2.7\cdot\cos(1.3t+0.9)\cdot(d)/(dt)(1.3t+0.9), \\ =1.3\cdot2.7\cdot\cos(1.3t+0.9), \\ =3.51\cdot\cos(1.3t+0.9). \end{gathered}`

Evaluating the function for t = 4 seconds, we get:

`v(4)=3.51\cdot\cos(1.3\cdot4+0.9)\cong3.451\Rightarrow v(4s)=3.451ft/s.`

b. The first instant after t = 0 when the velocity is 0 is obtained by solving for t the equation:

`\begin{gathered} v(t)=3.51\cdot\cos(1.3t+0.9)=0, \\ \cos(1.3t+0.9)=0, \\ 1.3t+0.9=(\pi)/(2), \\ 1.3t=(\pi)/(2)-0.9, \\ t=(1)/(1.3)\cdot((\pi)/(2)-0.9), \\ t\cong0.516, \\ t\cong0.516s. \end{gathered}`

c. The next instant in time when the velocity is 0 is obtained by solving for t the following equation:

`\begin{gathered} v(t)=3.51\cdot\cos(1.3t+0.9-\pi)=0, \\ \cos(1.3t+0.9-\pi)=0, \\ 1.3t+0.9-\pi=(\pi)/(2), \\ 1.3t=(\pi)/(2)-0.9+\pi, \\ t=(1)/(1.3)\cdot((3\pi)/(2)-0.9), \\ t\cong2.933, \\ t\cong2.933ft/s. \end{gathered}`

d. The acceleration of the object is obtained by taking the first derivative of the velocity:

`\begin{gathered} a(t)=v^(\prime)(t)=(d)/(dt)(3.51\cdot\cos(1.3t+0.9), \\ =3.51\cdot(d)/(dt)(\cos(1.3t+0.9)), \\ =3.51\cdot(-\sin(1.3t+0.9))\cdot(d)/(dt)(1.3t+0.9), \\ =3.51\cdot(-\sin(1.3t+0.9))\cdot1.3, \\ =-4.563\cdot\sin(1.3t+0.9). \end{gathered}`

When the height is h = 4, we have:

`\begin{gathered} h(t)=3.0+2.7\cdot\sin(1.3t+0.9)=4.0, \\ 2.7\cdot\sin(1.3t+0.9)=4.0-3.0, \\ 2.7\cdot\sin(1.3t+0.9)=1.0, \\ \sin(1.3t+0.9)=(1.0)/(2.7). \end{gathered}`

The acceleration when the height is h(t) = 1.0 is:

`\begin{gathered} a(t)=-4.563\cdot\sin(1.3t+0.9), \\ a(t)=-4.563\cdot(1.0)/(2.7), \\ a(t)=-1.690, \\ a(t)=-1.690ft/s^2. \end{gathered}`Answer

• a. 3.451 ft/s

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• b. 0.516 s

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• c. 2.933 s

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• d. -1.690 ft/s²