Q: A surveyor needs to determine the height of a building. Standing 50 meters from the base, she measures the angle of elevation to the top as $$38^\circ. $$What is the height of the building to the nearest meter?

$$h = 50 \cdot \tan(38^\circ) \approx 39$$ meters

Q: Which of the following is the correct conversion of the rectangular point $$(3, -3) to $$polar coordinates $$(r, \theta)$$, where $$\theta is in $$radians and $$-\pi \u003c \theta \leq \pi$$?

$$(r, \theta) = (3\sqrt{2}, -\dfrac{3\pi}{4})$$

Question 1

A surveyor needs to determine the height of a building. Standing 50 meters from the base, she measures the angle of elevation to the top as $$38^\circ. $$What is the height of the building to the nearest meter?

Accepted Answer

$$h = 50 \cdot 	an(38^\circ) \approx 39$$ meters

Question 2

Given triangle $$ABC$$ with $$A = 40^\circ$$, $$B = 65^\circ$$, and $$b = 12$$, what is the length of side $$a$$? (Use the Law of Sines.)

Accepted Answer

$$a = 12 \cdot \dfrac{\sin(40^\circ)}{\sin(65^\circ)} \approx 8.5$$

Question 3

Which of the following is the correct conversion of the rectangular point $$(3, -3) to $$polar coordinates $$(r, 	heta)$$, where $$	heta is in $$radians and $$-\pi \u003c 	heta \leq \pi$$?

Accepted Answer

$$(r, 	heta) = (3\sqrt{2}, -\dfrac{3\pi}{4})$$

Question 4

A triangle has sides $$a = 7$$, $$b = 9$$, and $$c = 12. $$What is the measure of angle $$C to $$the nearest degree? (Use the Law of Cosines.)

Accepted Answer

$$C = \cos$$^{-1}$$\left(\dfrac{7^2$ + 9^2$ - 12^2$}{2 \cdot 7 \cdot 9}\right) \approx 112^\circ$$

Question 5

Which of the following equations represents a circle centered at $$(2, -3)$$ with radius $$5$$?

Accepted Answer

$$(x - 2)^2$ + (y + 3)^2$ = 25$$

Question 6

Given the vector $$\vec{v} = \langle 6, 8 \rangle$$, what is the unit vector in the direction of $$\vec{v}$$?

Accepted Answer

$$\left\langle \dfrac{3}{5}, \dfrac{4}{5} \right\rangle$$

Question 7

Which of the following is the rectangular form of the complex number $$z = 4	ext{cis}\left(\dfrac{\pi}{3}ight)$$?

Accepted Answer

$$z = 2\sqrt{3} + 2i$$

Question 8

A force of $$20 N $$acts at an angle of $$30^\circ$$ above the horizontal. What are the horizontal and vertical components of the force?

Accepted Answer

Horizontal: $$20\cos(30^\circ) \approx 17.3 N$$, Vertical: $$20\sin(30^\circ) = 10 N$$

Question 9

Given the ellipse $$\dfrac{(x-1)^2$}{9} + \dfrac{(y+2)^2$}{4} = 1$$, what are the coordinates of its center and the lengths of its major and minor axes?

Accepted Answer

Center: $$(1, -2)$$; Major axis: $$6$$; Minor axis: $$4$$

Question 10

If $$\vec{a} = \langle 2, -1 \rangle$$ and $$\vec{b} = \langle -4, 2 \rangle$$, what is the dot product $$\vec{a} \cdot \vec{b}$$?

Accepted Answer

$$\vec{a} \cdot \vec{b} = (2)(-4) + (-1)(2) = -8 - 2 = -10$$

Question 11

Which of the following equations represents a parabola with vertex at $$(0,0)$$, focus at $$(0,2)$$, and directrix $$y = -2$$?

Accepted Answer

$$x^2 = 8y$

Question 12

Convert the polar equation $$r = 4\cos	heta to $$rectangular form.

Accepted Answer

$$x^2$$ + $$y^2 = 4x$

Question 13

Given the hyperbola $$\dfrac{(x-2)^2$}{9} - \dfrac{(y+1)^2$}{16} = 1$$, what are the coordinates of its center and the equations of its asymptotes?

Accepted Answer

Center: $$(2, -1)$$; Asymptotes: $$y + 1 = \pm \dfrac{4}{3}(x - 2)$$

Question 14

A pilot flies 200 km due east, then 150 km at $$30^\circ$$ north of east. What is the magnitude of the resultant displacement?

Accepted Answer

$$\sqrt{(200 + 150\cos 30^\$$circ)^2$$ + (150\sin 30^\$$circ)^2$$} \approx 336 km$$

Question 15

Which of the following statements is true about the vectors $$\vec{u} = \langle 3, 6 \rangle$$ and $$\vec{v} = \langle -2, 1 \rangle$$?

Accepted Answer

They are neither parallel nor orthogonal.

Question 16

Given the complex number $$z = -3 + 3i$$, what is its modulus and argument (in radians, $$-\pi \u003c 	heta \leq \pi$$)?

Accepted Answer

Modulus: $$3\sqrt{2}$$, Argument: $$\dfrac{3\pi}{4}$$

Question 17

A communications tower is supported by a cable attached to the ground 30 meters from the base. If the cable makes an angle of $$60^\circ$$ with the ground, how long is the cable?

Accepted Answer

$$\dfrac{30}{\cos 60^\circ} = 60$$ meters

Question 18

Which of the following is the correct result of multiplying the complex numbers $$z_1 = 2$$	ext{cis}(30^\circ)$$ and z_2$ = 3	ext{cis}(45^\circ) in $$polar form?

Accepted Answer

$$6	ext{cis}(75^\circ)$$

Question 19

Rewrite the equation $$x^2 + 4x$$ + $$y^2 - 6y = 12$ in standard form and identify the center and radius of the circle.

Accepted Answer

$$(x + 2)^2$ + (y - 3)^2$ = 25$$; Center: $$(-2, 3)$$; Radius: $$5$$

Precalculus Study Guide: Triangles, Polar Coordinates, Vectors, Complex Numbers, and Conic Sections

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