To solve for the horizontal distance from Kelly to the point directly below the kite and the height of the kite using 223 ft of string and the angle of elevation of 42 degrees, we can use trigonometric functions. Specifically, the sine and cosine functions will be useful here.
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Find the horizontal distance (\(d\)) from Kelly to the point directly below the kite. The cosine function relates the horizontal distance (adjacent side) to the hypotenuse (the length of the string):
\[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \]
So, we can express the horizontal distance as:
\[ d = \text{hypotenuse} \times \cos(\theta) \]
Substituting in the values:
\[ d = 223 \times \cos(42^\circ) \]
Calculating this:
\[ d \approx 223 \times 0.7431 \approx 165.72 , \text{ft} \]
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Find the height of the kite (\(h\)). The sine function relates the height (opposite side) to the hypotenuse:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]
So, we can express the height as:
\[ h = \text{hypotenuse} \times \sin(\theta) \]
Substituting in the values:
\[ h = 223 \times \sin(42^\circ) \]
Calculating this:
\[ h \approx 223 \times 0.6691 \approx 149.22 , \text{ft} \]
Summary of Results:
- The horizontal distance from Kelly to the point directly below the kite is approximately 165.72 ft.
- The height of the kite is approximately 149.22 ft.
From the options provided, both insights about the horizontal distance (165.72 ft) and kite height (149.22 ft) match the values stated.