# Parametric equations questions and answers Recent questions in Parametric equations, polar coordinates, and vector-valued functions
Parametric equations, polar coordinates, and vector-valued functions
ANSWERED ### Find an equation of the plane. The plane through the points (4, 1, 4), (5, -8, 6), and (-4, -5, 1)

Parametric equations, polar coordinates, and vector-valued functions
ANSWERED ### Find the exact length of the curve. Use a graph to determine the parameter interval. $$\displaystyle{r}={{\cos}^{{2}}{\left(\frac{\theta}{{2}}\right)}}$$

Parametric equations, polar coordinates, and vector-valued functions
ANSWERED ### The part of the surface $$\displaystyle{2}{y}+{4}{z}−{x}^{{2}}={5}$$ that lies above the triangle with vertices $$\displaystyle{\left({0},{0}\right)},{\left({2},{0}\right)},{\quad\text{and}\quad}{\left({2},{4}\right)}$$ Find the area of the surface.

Parametric equations, polar coordinates, and vector-valued functions
ANSWERED ### Find the points on the cone $$\displaystyle{z}^{{2}}={x}^{{2}}+{y}^{{2}}$$ that are closest tothe point (2, 2,0).

Parametric equations, polar coordinates, and vector-valued functions
ANSWERED ### Find a basis for the eigenspace corresponding to the eigenvalue of A given below. $$A=\begin{bmatrix}4 & 0&2&0 \\4 & 2&10&0\\3&-4&17&0\\2&-2&8&3 \end{bmatrix} , \lambda=3$$

Parametric equations, polar coordinates, and vector-valued functions
ANSWERED ### Find the area of the surface. The part of the paraboloid $$\displaystyle{z}={1}−{x}^{{2}}−{y}^{{2}}$$ that lies above the plane $$\displaystyle{z}=−{6}$$

Parametric equations, polar coordinates, and vector-valued functions
ANSWERED ### $$\displaystyle{x}={\sin{{\left({\frac{{\theta}}{{{2}}}}\right)}}},{y}={\cos{{\left({\frac{{\theta}}{{{2}}}}\right)}}},-\pi\leq\theta\leq\pi$$ (a) Eliminate the parameter to find a Cartesian equation of the curve. and how does thecurve look

Parametric equations, polar coordinates, and vector-valued functions
ANSWERED ### Find an equation of the following curve in polar coordinates and describe the curve. $$\displaystyle{x}={\left({1}+{\cos{{t}}}\right)}{\cos{{t}}}$$ $$\displaystyle{y}={\left({1}+{\cos{{t}}}\right)}{\sin{{t}}}.{0}\leq{t}\leq{2}\pi$$

Parametric equations, polar coordinates, and vector-valued functions
ANSWERED ### Find the length of the curve. $$\displaystyle{r}{\left({t}\right)}={\left\langle{6}{t},\ {t}^{{{2}}},\ {\frac{{{1}}}{{{9}}}}{t}^{{{3}}}\right\rangle},\ {0}\leq{t}\leq{1}$$

Parametric equations, polar coordinates, and vector-valued functions
ANSWERED ### Find a unit vector that is orthogonal to both $$\displaystyle{i}+{j}$$ and $$\displaystyle{i}+{k}$$.

Parametric equations, polar coordinates, and vector-valued functions
ANSWERED ### The following data was collected about students in Mr. Rexinger's high school statistics class. Wearing jeansNot wearing jeans Has long hair(below chin)77 Does not have long hair513 a. Mr. Rexinger is playing a game with his students. He randomly chooses a mystery student from his class roster. If a player guesses the hair length of the mystery student correctly, the player gets an early-lunch pass. Madeline is the next player. To have the greatest chance of winning an early-lunch pass, should she guess that the student has long hair? Explain. b. Mr. Rexinger tells Madeline that the mystery student is wearing jeans. Would you advise Madeline to change her guess? Explain. c. In a previous course, you may have studied the association of two numerical variables by analyzing scatterplots and least squares regression lines. Associations between categorical events, like having long hair or wearing jeans, are determined by independence-if two events are independent, then they are not associated. Are the events {not having long hair} and {wearing jeans} associated for the students in Mr. Rexinger's class today? Explain the independence relationship using P(A given B) = P(A). d. Are the events {not having long hair} and {wearing jeans} mutually exclusive? Explain.

Parametric equations, polar coordinates, and vector-valued functions
ANSWERED ### If f and g Are The Functions Whose Graphs Are Shown, Let u(x) = f(x)g(x) and $$\displaystyle{v}{\left({x}\right)}={\frac{{{f{{\left({x}\right)}}}}}{{{g{{\left({x}\right)}}}}}}$$ Parametric equations, polar coordinates, and vector-valued functions
ANSWERED ### The following data was collected about students in Mr. Rexinger's high school statistics class. Wearing jeansNot wearing jeans Has long hair(below chin)77 Does not have long hair513 a. Mr. Rexinger is playing a game with his students. He randomly chooses a mystery student from his class roster. If a player guesses the hair length of the mystery student correctly, the player gets an early-lunch pass. Madeline is the next player. To have the greatest chance of winning an early-lunch pass, should she guess that the student has long hair? Explain. b. Mr. Rexinger tells Madeline that the mystery student is wearing jeans. Would you advise Madeline to change her guess? Explain. c. In a previous course, you may have studied the association of two numerical variables by analyzing scatterplots and least squares regression lines. Associations between categorical events, like having long hair or wearing jeans, are determined by independence-if two events are independent, then they are not associated. Are the events {not having long hair} and {wearing jeans} associated for the students in Mr. Rexinger's class today? Explain the independence relationship using P(A given B) = P(A). d. Are the events {not having long hair} and {wearing jeans} mutually exclusive? Explain.

Parametric equations, polar coordinates, and vector-valued functions
ANSWERED ### Find the area of the parallelogram with vertices A(-3, 0), B(-1, 5), C(7, 4), and D(5, -1).

Parametric equations, polar coordinates, and vector-valued functions
ANSWERED ### Use the table of values of $$f(x, y)$$ to estimate the values of $$fx(3, 2)$$, $$fx(3, 2.2)$$, and $$fxy(3, 2)$$. $$\begin{array}{|c|c|}\hline y & 1.8 & 2.0 & 2.2 \\ \hline x & & & \\ \hline 2.5 & 12.5 & 10.2 & 9.3 \\ \hline 3.0 & 18.1 & 17.5 & 15.9 \\ \hline 3.5 & 20.0 & 22.4 & 26.1 \\ \hline \end{array}$$

Parametric equations, polar coordinates, and vector-valued functions
ANSWERED ### Determine whether the given vectors are orthogonal, parallel,or neither: $$u=(-3,\ 9,\ 6)$$ $$v=(4,\ -12,\ -8)$$

Parametric equations, polar coordinates, and vector-valued functions
ANSWERED ### Change from rectangular to cylindrical coordinates. (Let $$r\geq0$$ and $$0\leq\theta\leq2\pi$$.) a) $$(-2, 2, 2)$$ b) $$(-9,9\sqrt{3,6})$$ c) Use cylindrical coordinates. Evaluate $$\int\int\int_{E}xdV$$ where E is enclosed by the planes $$z=0$$ and $$z=x+y+10$$ and by the cylinders $$x^{2}+y^{2}=16$$ and $$x^{2}+y^{2}=36$$ d) Use cylindrical coordinates. Find the volume of the solid that is enclosed by the cone $$z=\sqrt{x^{2}+y^{2}}$$ and the sphere $$x^{2}+y^{2}+z^{2}=8$$.

Parametric equations, polar coordinates, and vector-valued functions
ANSWERED ### 1) Find the area of the part of the plane $$4x + 3y + z = 12$$ that lies in the first octant. 2) Use polar coordinates to find the volume of the given solid. Bounded by the paraboloid $$z = 5 + 2x^2 + 2y^2$$ and the plane z = 11 in the first octant

Parametric equations, polar coordinates, and vector-valued functions
ANSWERED ### Express as a trigonometric function of one angle. a) $$\cos2\sin(-9)-\cos9\sin2$$ Find the exact value of the expression. b) $$\sin\left(\arcsin\frac{\sqrt{3}}{2}+\arccos0\right)$$

Parametric equations, polar coordinates, and vector-valued functions
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