To find the distance between the ground and the kite, we'll use trigonometry, specifically the sine function. When drawing this scenario, we can visualize a right triangle formed by the kite string, the height of the kite above the ground, and the horizontal distance from the kite directly down to the ground.
Step-by-step explanation:
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Identify the Triangle:
- Consider the triangle formed by the kite string, the height of the kite above the ground, and the horizontal distance from where the kite is straight down to the ground.
- Label the angles and lengths:
- Let \( A \) be the point where the kite string is held on the ground.
- Let \( B \) be the position of the kite.
- Let \( C \) be the point directly underneath the kite on the ground.
- We form right triangle \( ABC \).
- The angle of elevation from point \( A \) to point \( B \) (the kite) is given as \( 40^\circ \).
- The length of the kite string \( AB \) is 100 feet (this serves as the hypotenuse of the triangle).
- Let \( h \) be the height from the horizontal ground to the kite (the vertical leg of the triangle, which is \( AC \)).
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Assign Variables:
- We know:
- Hypotenuse \( AB = 100 \) feet (length of the kite string)
- Angle \( \angle A = 40^\circ \)
- Height \( h = AC \)
- We know:
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Using the Sine Function:
- Sine is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right triangle: \[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]
- In this scenario: \[ \sin(40^\circ) = \frac{h}{100} \]
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Solve for Height \( h \):
- Rearranging the equation for \( h \): \[ h = 100 \cdot \sin(40^\circ) \]
- Next, calculate \( \sin(40^\circ) \):
- Using a calculator, we find that \( \sin(40^\circ) \approx 0.6428 \).
- Substitute this value into the equation: \[ h \approx 100 \cdot 0.6428 \approx 64.28 \]
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Rounding the Final Answer:
- When asked to round to the nearest foot: \[ h \approx 64 \text{ feet} \]
Conclusion:
The distance between the ground and the kite is approximately 64 feet. This represents the vertical height of the kite above the ground based on the given angle of elevation and length of the kite string.
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