A kite is attached to a 300 ft string, which makes a 42 degree angle with the level ground. The kite pilot is holding the string 4 ft above the ground.
a) How high above the ground is the kite?
b) Suppose there are power lines 200 feet in front of the kite flier that are 150 ft tall. Is there any risk that the kite will tangle in the power lines, if the kite flier doesn’t move?
(a) The height h above where it was being held is found via
h/300 = sin42°
(b) At 200 ft away, the height of the kite is
4 + 200 tan42°
or, compare 150^2 + 200^2 to the 300-ft length of the string. Is the string long enough to reach the wires?
To solve these problems, we can use trigonometry and the concept of right triangles. Let's begin!
a) To determine the height of the kite above the ground, we can use the trigonometric function tangent (tan).
First, let's draw a diagram to visualize the situation:
K
/ |
/ |h|
/ | |
/_________|
G d d
In the diagram, K represents the kite, G represents the ground, h represents the height of the kite, and d represents the horizontal distance between the kite and the ground.
We are given that the length of the string is 300 ft, and the angle between the string and the ground is 42 degrees. The distance between the string and the ground (d) is given as 4 ft.
Using the tangent function, we can set up the following equation:
tan(42 degrees) = h / d
Substituting the given values:
tan(42 degrees) = h / 4
To find h, we multiply both sides of the equation by 4:
h = 4 * tan(42 degrees)
Using a calculator, we can evaluate the expression:
h ≈ 4 * 0.9004
h ≈ 3.602 ft
Therefore, the kite is approximately 3.602 feet above the ground.
b) To determine if there is a risk of the kite tangling in the power lines, we need to compare the height of the power lines to the height of the kite.
We are given that the power lines are 200 ft in front of the kite flier and are 150 ft tall. The height of the kite above the ground is approximately 3.602 ft.
To check if the kite will tangle in the power lines, we need to consider the vertical distance between the kite and the power lines. This distance is given by:
Vertical distance = height of the power lines - height of the kite
Vertical distance = 150 ft - 3.602 ft
Vertical distance ≈ 146.398 ft
Since the vertical distance between the kite and the power lines is greater than zero, there is a risk of the kite tangling in the power lines if the kite flier doesn't move.
To avoid this risk, the kite flier should either move away from the power lines or adjust the height and angle of the string to ensure the kite remains below the height of the power lines.
To solve this problem, we can use trigonometric ratios and the given information to find the height of the kite and determine if there is a risk of it tangling in the power lines. Let's break it down step by step.
a) To find the height of the kite above the ground, we need to use the trigonometric tangent function (tan). The tangent of an angle is defined as the ratio of the opposite side to the adjacent side.
In our case:
- The angle formed between the string and the level ground is 42 degrees.
- The length of the string is 300 ft.
- The kite pilot is holding the string 4 ft above the ground.
Using the tangent function, we can say:
tan(42 degrees) = (opposite side) / (adjacent side)
Let's call the height of the kite "h". The opposite side is h + 4 ft (since the kite is held 4 ft above the ground), and the adjacent side is 300 ft.
Therefore, we have:
tan(42 degrees) = (h + 4) / 300
Now, we can solve this equation to find the value of h:
h = 300 * tan(42 degrees) - 4
Using a calculator, we can compute this and find the height of the kite above the ground.
b) To determine if there is any risk of the kite tangling in the power lines, we need to compare the height of the power lines with the height of the kite above the ground.
The power lines are located 200 ft in front of the kite flier and have a height of 150 ft. So, we need to check if the kite's height (h) is greater than the height of the power lines.
If h > 150 ft, there is a risk of the kite tangling in the power lines. If h <= 150 ft, there is no risk.
By calculating the value of h using the formula from part (a), we can compare it with the height of the power lines to determine if there is any risk.
Remember to consider the units (feet) while doing calculations and make sure to round the final answer to an appropriate level of precision.