Question

1. jada is standing 10 feet from the base of a tree and spots a nest sitting on a branch. the angle of elevation from the ground where she is standing to the nest is 55°. find the height of the nest.
2. the angle of elevation from the top of a 95-foot tall building to a hot air balloon in the sky is 76°. if the horizontal distance between the building and the hot air balloon is 354 feet, find the height of the hot air balloon.
3. a fire hydrant sits 72 feet from the base of a 125-foot tall building. find the angle of elevation from the fire hydrant to the top of the building.
4. a surfer is riding a 7-foot wave. the angle of depression from the surfer to the shoreline is 10°. what is the horizontal distance from the surfer to the shoreline?
5. a cell phone tower is anchored by two cables on each side for support. the cables stretch from the top of the tower to the ground, with each being equidistant from the base of the tower. the angle of depression from the top of the tower to the point in which the cable reaches the ground is 23°. if the tower is 140 feet tall, find the ground distance between the cables.

Explanation:

Problem 10

Step1: Define tangent ratio

$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

Step2: Substitute known values

$\tan(55^\circ) = \frac{x}{10}$

Step3: Solve for height $x$

$x = 10 \times \tan(55^\circ) \approx 10 \times 1.4281 = 14.28$

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Problem 11

Step1: Define tangent ratio

$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

Step2: Calculate balloon height above building

$h = 354 \times \tan(76^\circ) \approx 354 \times 4.0108 = 1419.82$

Step3: Total height of balloon

$\text{Total height} = 95 + 1419.82 = 1514.82$

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Problem 12

Step1: Define tangent ratio

$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

Step2: Substitute known values

$\tan(\theta) = \frac{125}{72} \approx 1.7361$

Step3: Solve for angle $\theta$

$\theta = \arctan(1.7361) \approx 60.1^\circ$

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Problem 13

Step1: Angle of depression = angle of elevation

$\theta = 10^\circ$, opposite side = 7 ft

Step2: Define tangent ratio

$\tan(10^\circ) = \frac{7}{d}$

Step3: Solve for horizontal distance $d$

$d = \frac{7}{\tan(10^\circ)} \approx \frac{7}{0.1763} = 39.70$

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Problem 14

Step1: Angle of depression = angle of elevation

$\theta = 23^\circ$, tower height = 140 ft

Step2: Define tangent ratio

$\tan(23^\circ) = \frac{140}{d}$

Step3: Solve for single cable distance

$d = \frac{140}{\tan(23^\circ)} \approx \frac{140}{0.4245} = 329.80$

Step4: Total distance between cables

$\text{Total distance} = 2 \times 329.80 = 659.60$

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Answer:

1. The height of the nest is approximately 14.3 feet
2. The height of the hot air balloon is approximately 1514.8 feet
3. The angle of elevation is approximately 60.1°
4. The horizontal distance is approximately 39.7 feet
5. The ground distance between the cables is approximately 660 feet