To solve this problem, we can use trigonometry. Let's break it down into smaller steps:
Step 1: Draw a diagram
First, let's draw a diagram to visualize the problem. We have a kite flying at a height of 100 meters above the ground, forming an angle of 60 degrees with the ground.
Step 2: Identify the relevant trigonometric ratio
Since we have the angle and the opposite side length (the height of the kite), we can use the sine function to solve this problem.
Step 3: Apply the sine function
According to the sine function, sin(angle) = opposite/hypotenuse. In this case, the opposite side is the height of the kite (100 meters) and the hypotenuse is the length of the kite string that we need to find.
So, sin(60 degrees) = 100/hypotenuse.
Step 4: Solve for the hypotenuse
To solve for the hypotenuse, we need to isolate it in the equation. We can do this by multiplying both sides of the equation by the hypotenuse:
hypotenuse * sin(60 degrees) = 100.
Step 5: Calculate the hypotenuse
To find the hypotenuse, we need to evaluate sin(60 degrees). The sine of 60 degrees is √3/2.
So, hypotenuse * (√3/2) = 100.
Now, we can solve for the hypotenuse:
hypotenuse = 100 / (√3/2).
hypotenuse = 100 * (2/√3).
hypotenuse = (200/√3).
hypotenuse = (200/√3) * (√3/√3).
hypotenuse = (200√3) / 3.
Therefore, the length of the kite string is (200√3) / 3 meters.
Step 6: Simplify the answer
To simplify the answer, we can multiply both the numerator and denominator by √3:
(200√3) / 3 = (200√3 * √3) / (3 * √3).
(200√3) / 3 = (200 * 3) / (3√3).
(200√3) / 3 = 600 / (3√3).
(200√3) / 3 = 200 / √3.
Therefore, the length of the kite string is 200 / √3 meters.
Step 7: Rationalize the denominator
To rationalize the denominator, we can multiply both the numerator and denominator by √3:
(200 / √3) * (√3 / √3) = (200√3) / 3.
Therefore, the length of the kite string is (200√3) / 3 meters.
So, the correct answer is option c) \(25 \sqrt{3}\) meters.
Answers & Comments
To solve this problem, we can use trigonometry. Let's break it down into smaller steps:
Step 1: Draw a diagram
First, let's draw a diagram to visualize the problem. We have a kite flying at a height of 100 meters above the ground, forming an angle of 60 degrees with the ground.
Step 2: Identify the relevant trigonometric ratio
Since we have the angle and the opposite side length (the height of the kite), we can use the sine function to solve this problem.
Step 3: Apply the sine function
According to the sine function, sin(angle) = opposite/hypotenuse. In this case, the opposite side is the height of the kite (100 meters) and the hypotenuse is the length of the kite string that we need to find.
So, sin(60 degrees) = 100/hypotenuse.
Step 4: Solve for the hypotenuse
To solve for the hypotenuse, we need to isolate it in the equation. We can do this by multiplying both sides of the equation by the hypotenuse:
hypotenuse * sin(60 degrees) = 100.
Step 5: Calculate the hypotenuse
To find the hypotenuse, we need to evaluate sin(60 degrees). The sine of 60 degrees is √3/2.
So, hypotenuse * (√3/2) = 100.
Now, we can solve for the hypotenuse:
hypotenuse = 100 / (√3/2).
hypotenuse = 100 * (2/√3).
hypotenuse = (200/√3).
hypotenuse = (200/√3) * (√3/√3).
hypotenuse = (200√3) / 3.
Therefore, the length of the kite string is (200√3) / 3 meters.
Step 6: Simplify the answer
To simplify the answer, we can multiply both the numerator and denominator by √3:
(200√3) / 3 = (200√3 * √3) / (3 * √3).
(200√3) / 3 = (200 * 3) / (3√3).
(200√3) / 3 = 600 / (3√3).
(200√3) / 3 = 200 / √3.
Therefore, the length of the kite string is 200 / √3 meters.
Step 7: Rationalize the denominator
To rationalize the denominator, we can multiply both the numerator and denominator by √3:
(200 / √3) * (√3 / √3) = (200√3) / 3.
Therefore, the length of the kite string is (200√3) / 3 meters.
So, the correct answer is option c) \(25 \sqrt{3}\) meters.