Question

1. (05.02 mc)

quinn is flying a kite. the angle of elevation formed by the kite string and the ground is 46°, and the kite string forms a straight segment that is 80 feet long.

explain how to find the distance between the ground and the kite. include a description of the triangle you drew to help you solve, including the variables and measurements you assigned to each side and angle. round your answer to the nearest foot. (10 points)

1. (05.03 mc)

an observer (o) is located 400 feet from a building (b). the observer notices a kite (k) flying at a 29° angle of elevation from his line of sight. how high is the kite flying over the building? you must show all work and calculations to receive full credit. (10 points)

Explanation:

Model the first scenario

Using the Right Triangle Trigonometry and Angle of Elevation Applications knowledge points

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Calculate the kite's height

Using the Sine Ratio knowledge point

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Model the second scenario

Using the Right Triangle Trigonometry and Angle of Elevation Applications knowledge points

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Calculate the height over the building

Using the Tangent Ratio knowledge point

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Answer:

Question 3

To find the distance between the ground and the kite, model the scenario using a right triangle \(\triangle ABC\), where:

• Vertex \(A\) represents the person on the ground, with the angle of elevation \(\angle A = 46^\circ\).
• Vertex \(B\) represents the kite in the air.
• Vertex \(C\) represents the point on the ground directly below the kite, forming a right angle \(\angle C = 90^\circ\).
• The hypotenuse \(AB\) represents the kite string, with a length of \(80\text{ feet}\).
• The side opposite to \(\angle A\), which is \(BC\) (labeled as \(h\)), represents the vertical height of the kite above the ground.

Using the sine ratio:

$$\sin(46^\circ) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{h}{80}$$

Solving for \(h\):

$$h = 80 \cdot \sin(46^\circ) \approx 58\text{ feet}$$

The distance between the ground and the kite is approximately \(58\text{ feet}\).

Question 4

To find how high the kite is flying over the building, model the scenario with right triangle \(\triangle OBK\), where:

• \(O\) is the observer's position.
• \(B\) is the base of the building, with \(\angle B = 90^\circ\).
• \(K\) is the position of the kite directly above the building.
• The adjacent side \(OB = 400\text{ feet}\).
• The angle of elevation \(\angle O = 29^\circ\).
• The opposite side \(BK = H\) represents the height of the kite.

Using the tangent ratio:

$$\tan(29^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{H}{400}$$

Solving for \(H\):

$$H = 400 \cdot \tan(29^\circ) \approx 222\text{ feet}$$

The kite is flying approximately \(222\text{ feet}\) over the building.