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):
[tex]h(t)=3.0+2.7\cdot\sin(1.3t+0.9).[/tex]a. The velocity of the object is obtained by taking the first derivative:
[tex]\begin{gathered} v(t)=h^{\prime}(t)=\frac{d}{dt}(3.0+2.7\cdot\sin(1.3t+0.9)) \\ =0+2.7\cdot\frac{d}{dt}\sin(1.3t+0.9)), \\ =2.7\cdot\cos(1.3t+0.9)\cdot\frac{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}[/tex]Evaluating the function for t = 4 seconds, we get:
[tex]v(4)=3.51\cdot\cos(1.3\cdot4+0.9)\cong3.451\Rightarrow v(4s)=3.451ft/s.[/tex]b. The first instant after t = 0 when the velocity is 0 is obtained by solving for t the equation:
[tex]\begin{gathered} v(t)=3.51\cdot\cos(1.3t+0.9)=0, \\ \cos(1.3t+0.9)=0, \\ 1.3t+0.9=\frac{\pi}{2}, \\ 1.3t=\frac{\pi}{2}-0.9, \\ t=\frac{1}{1.3}\cdot(\frac{\pi}{2}-0.9), \\ t\cong0.516, \\ t\cong0.516s. \end{gathered}[/tex]c. The next instant in time when the velocity is 0 is obtained by solving for t the following equation:
[tex]\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=\frac{\pi}{2}, \\ 1.3t=\frac{\pi}{2}-0.9+\pi, \\ t=\frac{1}{1.3}\cdot(\frac{3\pi}{2}-0.9), \\ t\cong2.933, \\ t\cong2.933ft/s. \end{gathered}[/tex]d. The acceleration of the object is obtained by taking the first derivative of the velocity:
[tex]\begin{gathered} a(t)=v^{\prime}(t)=\frac{d}{dt}(3.51\cdot\cos(1.3t+0.9), \\ =3.51\cdot\frac{d}{dt}(\cos(1.3t+0.9)), \\ =3.51\cdot(-\sin(1.3t+0.9))\cdot\frac{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}[/tex]When the height is h = 4, we have:
[tex]\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)=\frac{1.0}{2.7}. \end{gathered}[/tex]The acceleration when the height is h(t) = 1.0 is:
[tex]\begin{gathered} a(t)=-4.563\cdot\sin(1.3t+0.9), \\ a(t)=-4.563\cdot\frac{1.0}{2.7}, \\ a(t)=-1.690, \\ a(t)=-1.690ft/s^2. \end{gathered}[/tex]Answer• a. 3.451 ft/s
,• b. 0.516 s
,• c. 2.933 s
,• d. -1.690 ft/s²
