Respuesta :
Answer:
1) The velocity at this time is of [tex]-9\frac{\sqrt{3}}{2}[/tex] meters per second.
2) The velocity at this time is of [tex]13\frac{\sqrt{2}}{2}[/tex] meters per second.
Step-by-step explanation:
This question involves concepts of derivatives.
The velocity is the derivative of the position.
We use these following derivatives:
[tex](\sin{x})^{\prime} = \cos{x}[/tex]
[tex](\cos{x})^{\prime} = -\sin{x}[/tex]
1) s = 1 + 9(cos t)
Find the body's speed at time t=pi/3 sec.
We have to find the derivative at [tex]t = \frac{\pi}{3}[/tex]. So
[tex]v = (1 + 9\cos{t})^{\prime} = -9\sin{t}[/tex]
[tex]v(\frac{\pi}{3}) = -9\sin{\frac{\pi}{3}}[/tex]
[tex]\frac{\pi}{3}[/tex] is a common angle, which has a sine of [tex]\frac{\sqrt{3}}{2}[/tex]. So
[tex]-9\sin{\frac{\pi}{3}} = -9\frac{\sqrt{3}}{2}[/tex]
The velocity at this time is of [tex]-9\frac{\sqrt{3}}{2}[/tex] meters per second.
2) s = 12(sin t) - (cos t)
Find the body's velocity at time t=pi/4 sec.
We have to find the derivative at [tex]t = \frac{\pi}{4}[/tex]. So
[tex]v = (12\sin{t} - \cos{t})^{\prime} = 12\cos{t} + \sin{t}[/tex]
[tex]\frac{\pi}{4}[/tex] is a common angle, which has both sine and cosine of [tex]\frac{\sqrt{2}}{2}[/tex]. So
[tex]12\cos{\frac{\pi}{4}} + \sin{\frac{\pi}{4}} = 12\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} = 13\frac{\sqrt{2}}{2}[/tex]
The velocity at this time is of [tex]13\frac{\sqrt{2}}{2}[/tex] meters per second.
