Question

37.

x =

y =

38.

x =

y =

39.

x =

y =

40.

x =

y =

41.

x =

y =

42.

x =

y =

z =

43.

x =

y =

z =

44.

x =

y =

45.

x =

y =

options: (2sqrt{2}), (3sqrt{3}), (\frac{4sqrt{3}}{3}), (6), (6sqrt{3}), (8sqrt{2}), (12), (12sqrt{2}), (16), (\frac{16sqrt{3}}{3}), (17sqrt{3}), (18sqrt{15}), (22sqrt{2}), (22sqrt{6}), (34), (36sqrt{5}), (44)

Explanation:

Problem 37

Step1: Find x via 30-60-90 ratios

$x = 17\sqrt{3}$

Step2: Find y via 30-60-90 ratios

$y = 34$

Problem 38

(Answers already provided as $x=12$, $y=12\sqrt{2}$)

Problem 39

Step1: Solve for x using tan(30°)

$\tan(30^\circ) = \frac{x}{18\sqrt{5}} \implies x = 18\sqrt{5} \times \frac{1}{\sqrt{3}} = 6\sqrt{15}$

Step2: Solve for y using cos(30°)

$\cos(30^\circ) = \frac{18\sqrt{5}}{y} \implies y = \frac{18\sqrt{5}}{\frac{\sqrt{3}}{2}} = 12\sqrt{15}$

Problem 40

Step1: Find x via 30-60-90 triangle

$x = 12 \times \frac{\sqrt{3}}{2} = 6\sqrt{3}$

Step2: Find y via 30-60-90 triangle

$y = 12 \times \frac{1}{2} = 6$

Problem 41

Step1: Find y via 30-60-90 ratios

$y = 6 \times 2 = 12$

Step2: Find x via 30-60-90 ratios

$x = 6\sqrt{3}$

Problem 42

Step1: Find x via 45-45-90 triangle

$x = 2\sqrt{2}$

Step2: Find y via 30-60-90 triangle

$y = 2 \times 2 = 4$

Step3: Find z via Pythagorean theorem

$z = \sqrt{(4+2)^2 + (6\sqrt{3})^2} = \sqrt{36 + 108} = \sqrt{144} = 12$

Problem 43

Step1: Find y via 45-45-90 triangle

$y = 44 \times \frac{\sqrt{2}}{2} = 22\sqrt{2}$

Step2: Find x via sine of 60°

$\sin(60^\circ) = \frac{22\sqrt{2}}{x} \implies x = \frac{22\sqrt{2}}{\frac{\sqrt{3}}{2}} = \frac{44\sqrt{6}}{3}$

Step3: Find z via cosine of 60°

$\cos(60^\circ) = \frac{z}{x} \implies z = \frac{44\sqrt{6}}{3} \times \frac{1}{2} = \frac{22\sqrt{6}}{3}$

Problem 44

(Answers already provided as $x=8\sqrt{2}$, $y=16$)

Problem 45

Step1: Find x via sine of 60°

$\sin(60^\circ) = \frac{12}{x} \implies x = \frac{12}{\frac{\sqrt{3}}{2}} = 8\sqrt{3}$

Step2: (y already provided as 12)

Answer:

1. $x=17\sqrt{3}$, $y=34$
2. $x=12$, $y=12\sqrt{2}$
3. $x=6\sqrt{15}$, $y=12\sqrt{15}$
4. $x=6\sqrt{3}$, $y=6$
5. $x=6\sqrt{3}$, $y=12$
6. $x=2\sqrt{2}$, $y=4$, $z=12$
7. $x=\frac{44\sqrt{6}}{3}$, $y=22\sqrt{2}$, $z=\frac{22\sqrt{6}}{3}$
8. $x=8\sqrt{2}$, $y=16$
9. $x=8\sqrt{3}$, $y=12$