Exam 14: Parametric Equations and Conic Sections

Is the function $4 x^{3}+4=y$ implicit or explicit?

Is the equation $y=x^{2}+4$ explicit or implicit?

A search light uses a $6 \mathrm{ft}$ wide parabolic mirror that is $4 \mathrm{ft}$ deep. If the lamp is positioned $0.141 \mathrm{ft}$ above the center of the mirror, how wide is the beam in the sky?

Could the graph below be the graph of the parametric equations $x=-t+3, y=\sqrt{t}$ ? Assume the parameter is restricted to values for which the functions are defined.

Describe the graph of the curve $x=t^{4}+6, y=t^{3}+7$ .

A satellite moves in an elliptical orbit with the moon at a focus. The nearest it gets to the moon is 1.21 million $\mathrm{km}$ and the farthest is 4.1 million $\mathrm{km}$ . What is the length of the major and minor axes? Round to 2 decimal places.

A mouse hanging on to the end of the windmill blade shown below has coordinates given by $x=A \cos (k t), y=A \sin (k t)$ , where the origin is at the center of the blades, $x$ and $y$ are in meters, and $t$ is in seconds. The blades are 5 meters long and the windmill makes one complete revolution every 28 seconds in a counterclockwise direction. The mouse starts in the 3 o'clock position and, 3.5 seconds later, loses its hold and flies off. How many degrees is the angle between the blade and the positive $x$ -axis at the moment the mouse flies off?

Write a parameterization for one branch of the following hyperbola using hyperbolic functions: $\frac{(x-2)^{2}}{4}-\frac{(y+5)^{2}}{36}=1, x<2$

Each end of a string is tacked to the point $(4,0)$ and $(-4,0)$ on inch graph paper. The string is 12 inches long. A pencil pulls the string taut and rotates to sweep out an ellipse. What are the focal points of the ellipse?

The directrix of the parabola $2 y=x^{2}-14 x+53$ is at $y=$ -----------.

Let $\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1$ be the equation of the ellipse shown below. Assuming that the $x$ - and $y$ -scales are equal, which of the following is true?

Are the parametric equations for the quarter of an ellipse centered at $(0,0)$ , starting at $(0,2)$ and ending at $(-4,0)$ , given by $x=4 \cos t, y=2 \sin t, \frac{\pi}{2} \leq t \leq \pi$ ?

Write $\cosh (6 x) \sinh (7 x)$ in terms of powers of $e$ .

Select the type of curve formed by the parametric equations $x=4+\cos ^{2} t$ and $y=-3-\sin t$ .

Let $x=a+k \cos t$ and $y=b+k \sin t$ parameterize a circle, with $a<0, b<0, k<a$ , and $k<b$ . What quadrant does the circle lie in?

The ellipse given by the equations $x=8+4 \cos t, y=4+5 \sin t, 0 \leq t \leq 2 \pi$ , has a minor axis of length ------------- units.

Select the type of curve formed by the parametric equations $x=2+3 \cos t$ and $y=2+3 \sin t$ .

What is the radius of the circle parameterized by $x=9 \cos t, y=9 \sin t, 0 \leq t \leq 2 \pi$ ?

As $t$ varies, the equations $x=2+6 \cos t$ and $y=-3+6 \sin t$ trace out a circle. a) Describe the center and radius of the circle. b) If $0 \leq t \leq \frac{\pi}{2}$ , what part of the circle is obtained?

What is the radius of the circle $5(x-5)^{2}+4(y+6)^{2}-45=-(y+6)^{2}$ ?