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Quinn is flying a kite. The angle of elevation formed by the kite string and the ground is 40°, and the kite string forms a straight segment that is 100 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 answer

To find the distance between the ground and the kite, we can model the situation using a right triangle. Here’s how to break it down step by step:

  1. Understanding the Triangle:

    • We have a right triangle where:
      • One vertex is at the point where Quinn is standing on the ground (point A).
      • The second vertex is at the kite in the air (point B).
      • The third vertex is directly below the kite on the ground (point C).
    • The angle of elevation from the ground to the kite string is 40° (angle A), which is the angle formed between the ground (the horizontal side AC) and the kite string (side AB).
    • The length of the kite string (hypotenuse AB) is given as 100 feet.

  2. Identifying the Sides:

    • AB is the kite string (hypotenuse) = 100 feet.
    • AC is the horizontal distance from Quinn to the point directly below the kite on the ground.
    • BC is the vertical distance from the ground to the kite (this is the height we want to find).

  3. Using Trigonometric Functions:

    • In a right triangle, we can use the sine function to find the height (BC) since we know the hypotenuse (AB) and the angle (A). The sine of an angle in a right triangle is defined as the ratio of the opposite side to the hypotenuse: \[ \sin(A) = \frac{\text{opposite}}{\text{hypotenuse}} \] In our triangle, the opposite side is BC (the height), and the hypotenuse is AB.

  4. Setting Up the Equation:

    • Plug in the known values: \[ \sin(40°) = \frac{BC}{100} \]

  5. Solving for BC:

    • Multiply both sides of the equation by 100: \[ BC = 100 \cdot \sin(40°) \]

  6. Calculating the Value:

    • Now we need to calculate the sine of 40°: \[ \sin(40°) \approx 0.6428 \quad (\text{using a calculator or a sine table}) \]
    • Substitute this value into the equation: \[ BC \approx 100 \cdot 0.6428 \approx 64.28 \]
    • Rounding to the nearest foot gives us: \[ BC \approx 64 \text{ feet} \]

  7. Conclusion:

    • The distance from the ground to the kite, represented by BC, is approximately 64 feet.

In summary, by using trigonometry and the given angle of elevation, we constructed a right triangle and used the sine function to find that the height of the kite is around 64 feet.