Answer
(a) radius=$3$. $(0,3)$ at t=0, clockwise, $p=\pi$,
(b) $x=3sin(4t), y=3cos(4t)$
(c) $x^2+y^2=9$
(d) $r=3$
Work Step by Step
(a) The parametric equations represent a circle centered at $(0,0)$ with a radius of $3$. At $t=0$, the object is at $(0,3)$, at $t=\frac{\pi}{4}$, it moves to $(3,0)$ indicating it is moving clockwise. The period of the function is $p=\frac{2\pi}{2}=\pi$, so it takes the object $\pi$ seconds for one revolution around the circle.
(b) When the speed of the object doubles, the period will be half of the original, so $p'=p/2=\pi/2$, so the new set of equations are $x=3sin(4t), y=3cos(4t)$
(c) Take the square and sum up the original set of equations, we have $x^2+y^2=9sin^2(2t)+9cos^2(2t)=9$
(Pythagorean Theorem). The rectangular equation $x^2+y^2=9$
(d) Use the formula $r^2=x^2+y^2$, the above equation becomes $r^2=9$ or $r=3$ ($r=-3$ is the same). The polar equation $r=3$
