The values of x and y at t=3 are -4.95 and 0.423 respectively.
The value of dx/dt and dy/dt are 0.71 and -2.97 respectively.
The value of the tangent slope at t=3 is 4.21.
The speed at t=3 is 3.05 units/sec.
Given the equation of x as the function of t is
x= 5 cos t
similarly, the equation of y as the function of t is
y= 3 sin t
At t=3 the value of x will be
x (at t=3) = 5 cos 3= 5(-0.989)= -4.95
At t=3 the value of y will be
y (at t=3) = 3 sin 3= 3(0.141)= 0.423
The derivative of the function of x with respect to t will be
dx/dt= d(5 cos t)/dt= 5d(cos t)/dt= -5 sin t
at t=3 the value of dx/dt will be
dx/dt (at t=3) = -5 sin 3= -5(0.141)= 0.71
The derivative of the function of y with respect to t will be
dy/dt= d(3 sin t)/dt= 3d(sin t)/dt= 3 cos t
at t=3 the value of dy/dt will be
dy/dt (at t=3) = 3 cos t= 3(-0.989)= -2.97
The tangent slope is dy/dx which can be calculated by
dy/dx= (dy/dt)(dt/dx)= (dy/dt)/(dx/dt)= 3 cos t/ -5 sin t= (-3/5) cot t
at t=3 the value of tangent slope will be
dy/dx (at t=3) = (-3/5) cot 3= 4.21
The speed at t=3 will be
speed v= [tex]\sqrt{v_{x} ^{2} + v_{y} ^{2} }[/tex]
= √(dx/dt)²+(dy/dt)²
at t=3
= √(0.71)²+(-2.97)²
= √9.325
= 3.05 unit/sec
Therefore the values of x and y at t=3 are -4.95 and 0.423 respectively.
The value of dx/dt and dy/dt are 0.71 and -2.97 respectively.
The value of the tangent slope at t=3 is 4.21.
The speed at t=3 is 3.05 units/sec.
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