Q: Write the equation $$x^2$$ + $$4y^2 = 4$ using polar coordinates (r$$, \theta)$$. Which is correct?

$$r^2$$\$$cos^2$$\theta + $$4r^2$$\$$sin^2$$\theta = 4$$

Question 1

A 25-foot ladder just reaches the top of a house and forms an angle of $$41.5^{\circ}$$ with the ground. How tall is the house? (Round to the nearest 0.1 foot.)

Accepted Answer

$$18.7 ft$$

Question 2

A surveyor measures angle $$DAB = 25$$^{\circ}$$, walks L = 1950$ feet to point C$, and measures angle $$ACB = $$15^{\circ}. $$What is the distance from $$A to$$ $$B$$? (Round to two decimal places.)

Accepted Answer

$$2906.43 ft$$

Question 3

Convert the polar coordinates $$\left(3, \dfrac{3\pi}{4}\right) to $$rectangular coordinates.

Accepted Answer

$$\left(-\dfrac{3\sqrt{2}}{2}, \dfrac{3\sqrt{2}}{2}\right)$$

Question 4

Write the equation $$x^2$$ + $$4y^2 = 4$ using polar coordinates (r$$, 	heta)$$. Which is correct?

Accepted Answer

$$r^2$$\$$cos^2$$	heta + $$4r^2$$\$$sin^2$$	heta = 4$$

Question 5

Convert the polar equation $$r = \dfrac{5}{1 + \cos	heta} to $$rectangular coordinates.

Accepted Answer

$$y^2 = 25 - 10x$

Question 6

Find polar coordinates $$(r, 	heta)$$ for the point $$(5, -5)$$, where $$r \u003e 0$$ and $$-\pi \u003c 	heta \leq \pi. $$Express $$	heta in $$radians.

Accepted Answer

$$\left(5\sqrt{2}, -\dfrac{\pi}{4}\right)$$

Question 7

Transform $$	heta = \dfrac{7\pi}{6} to $$rectangular form. What is the slope-intercept form of the equation?

Accepted Answer

$$y = \dfrac{1}{\sqrt{3}}x$$

Question 8

Write $$-3i in $$polar form, expressing the argument in degrees.

Accepted Answer

$$3(\cos 270$$^{\circ}$$ + i\sin 270$$^{\circ}$$)$$

Question 9

Given $$z = 5(\cos 35$$^{\circ}$$ + i\sin 35$$^{\circ}$$)$$ and $$w = 2(\cos 40$$^{\circ}$$ + i\sin 40$$^{\circ}$$)$$, find $$zw in $$polar form.

Accepted Answer

$$10(\cos 75$$^{\circ}$$ + i\sin 75$$^{\circ}$$)$$

Question 10

Given $$z = 10(\cos 45$$^{\circ}$$ + i\sin 45$$^{\circ}$$)$$ and $$w = 5(\cos 15$$^{\circ}$$ + i\sin 15$$^{\circ}$$)$$, find $$\dfrac{z}{w} in $$polar form.

Accepted Answer

$$2(\cos 30$$^{\circ}$$ + i\sin 30$$^{\circ}$$)$$

Question 11

Write $$4\left(\cos\dfrac{3\pi}{4} + i\sin\dfrac{3\pi}{4}\right) in $$standard form $$a + bi.$$

Accepted Answer

$$-4$$

Question 12

If $$\vec{v} = -6\mathbf{i} + 6\mathbf{j}$$ and $$\vec{w} = -3\mathbf{i} + 5\mathbf{j}$$, what is $$\vec{v} - \vec{w}$$?

Accepted Answer

$$-3\mathbf{i} + \mathbf{j}$$

Question 13

Find the direction angle of $$\vec{v} = 8\mathbf{i} - 7\mathbf{j}. ($$Round to one decimal place.)

Accepted Answer

$$318.8^{\circ}$$

Question 14

Given $$\vec{u} = 2x\mathbf{i} + 7\mathbf{j}$$ and $$\vec{v} = x\mathbf{i} - 8\mathbf{j}$$, find all $$x so $$that $$\vec{u}$$ and $$\vec{v}$$ are orthogonal.

Accepted Answer

$$x = 2,\ -2$$

Question 15

Write the equation for the ellipse centered at $$(1, -2)$$ with $$a = 3 ($$horizontal axis) and $$b = 2 ($$vertical axis).

Accepted Answer

$$\dfrac{(x-1)^2$}{9} + \dfrac{(y+2)^2$}{4} = 1$$

Question 16

Given the ellipse $$\dfrac{x^2$}{25} + \dfrac{y^2$}{49} = 1$$, what are the center, foci, and vertices?

Accepted Answer

center at $$(0,0)$$; foci at $$(0, -2\sqrt{6})$$ and $$(0, 2\sqrt{6})$$; vertices at $$(0, -7)$$ and $$(0, 7)$$

Question 17

Find the vertex, focus, and directrix of the parabola $$x = -20y^2$.

Accepted Answer

Vertex: $$(0,0)$$; Focus: $$(0, -5)$$; Directrix: $$y = 5$$

Question 18

Which of the following is the correct graph for the hyperbola $$36y^2$$ - $$9x^2 = 324$?

Accepted Answer

A hyperbola opening up and down (vertical transverse axis)

Question 19

Find parametric equations for the circle of radius $$4$$ centered at the origin, traversed counterclockwise starting and ending at $$(4,0)$$ for $$0 \leq t \leq 2\pi.$$

Accepted Answer

$$x = 4\cos t,\ y = 4\sin t$$

Question 20

Find parametric equations for an object moving clockwise along the ellipse $$\dfrac{x^2$}{9} + \dfrac{y^2$}{16} = 1$$, beginning at $$(3,0)$$ and requiring $$8$$ seconds for a complete revolution.

Accepted Answer

$$x = 3\cos\left(-\dfrac{t\pi}{4}\right),\ y = 4\sin\left(-\dfrac{t\pi}{4}\right)$$

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