Question

Consider an object moving in the plane whose location at time t seconds is given by the parametric equations: x(t)=4cos(πt) y(t)=3sin(πt). Assume the distance units in the plane are meters. (a) The object is moving around an ellipse (as in the previous problem) with equation: x2 a2 + y2 b2 =

Answer 1

Answer:

Since the parametric equation is given as

$x(t)=4cos(\pt t),y(t)=4sin(\pi t)\\\\\therefore x(t)^{2}+y(t)^{2}=16cos^{2}(\pi t)+16sin^{2}(\pi t)\\\\x(t)^{2}+y(t)^{2}=16$

$\therefore \frac{x^{2}}{4^{2}}+\frac{y^{2}}{4^{2}}=1\\\\comparingwith\\\\\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\\\\\therefore a=4,b=4$

Answer 2

x(t)=4cos(πt) -divide by 4

y(t)=3sin(πt) -divide by 3

x(t)/4 = cos(πt) -to the square

y(t)/3 = sin(πt) -to the square

x(t)²/16 = cos²(πt)

y(t)²/9 = sin²(πt)

Sum this:

x(t)²/16 + y(t)²/9 = cos²(πt)+sin²(πt)

x²/16 + y²/9 = 1.

a² is 16 and b² is 9 therefore a=4 and b=3.

Parametric equation of elipse is generally:

x(t) = a cos(f(t))

y(t) = b sin (f(t))