To find the distance between the ground and the kite, we can visualize the situation as a right triangle.
Here's how to break it down:
-
Identify the Components of the Triangle:
- The right triangle is formed by the kite string, the vertical distance from the kite to the ground, and the horizontal distance from the point directly below the kite to the point where Quinn is holding the kite string.
- The angle of elevation from the ground to the kite string is 40°.
- The length of the kite string represents the hypotenuse of the right triangle, which is 100 feet.
-
Label the Triangle:
- Let \( A \) be the point where Quinn is holding the kite string (the base).
- Let \( B \) be the point where the kite is located (the top).
- Let \( C \) be the point directly below the kite on the ground.
- Therefore, the triangle \( ABC \) is a right triangle where:
- \( AB \) is the hypotenuse (length of the kite string = 100 ft).
- \( AC \) is the horizontal distance from the point on the ground directly below the kite to where Quinn is standing (we do not need this for our calculation).
- \( BC \) is the vertical distance from the kite down to the ground (this is what we're trying to find, which we will call \( h \)).
-
Using Trigonometry:
- In a right triangle, the sine of an angle is the ratio of the length of the opposite side (in this case, \( BC \)) to the hypotenuse (\( AB \)).
- The angle we have is 40°, and we can express this relationship mathematically: \[ \sin(40°) = \frac{BC}{AB} \] Substituting the known values: \[ \sin(40°) = \frac{h}{100} \] This can be rearranged to solve for the height \( h \): \[ h = 100 \cdot \sin(40°) \]
-
Calculating the Height:
- Now, we will calculate \( h \): \[ h \approx 100 \cdot \sin(40°) \] Using a calculator, we find: \[ \sin(40°) \approx 0.6428 \] Therefore: \[ h \approx 100 \cdot 0.6428 = 64.28 \]
-
Round to the Nearest Foot:
- Rounding \( 64.28 \) to the nearest foot gives us \( 64 \).
Thus, the distance between the ground and the kite is approximately 64 feet.