Final answer:
Using the Pythagorean theorem, the kite's string length is determined to be 291.55 m when the kite is 250 m away horizontally. The speed of the string being let out is found by differentiating the string length with respect to time, resulting in an approximate speed of 8.57 m/s.
Step-by-step explanation:
The question concerns the rate at which the kite's string is being released, given that the kite is flying horizontally at a certain height and speed. To solve this, we need to apply the Pythagorean theorem. We know the kite is 151.5 meters above the ground, but the boy's height is 1.5 meters, therefore the vertical distance we will use for the kite from the boy's hands is 150 meters. Given that the kite is 250 meters away from the boy, we can find the length of the string by using the Pythagorean theorem:
L² = a² + b²
where L is the length of the string, a = 250 m is the horizontal distance, and b = 150 m is the vertical distance. Therefore:
L² = (250 m)² + (150 m)²
L² = 62500 m² + 22500 m²
L² = 85000 m²
L = √(85000 m²)
L = 291.55 m
To find out how fast the string is being let out, we need to consider the speed of the kite. The horizontal component of the string's speed is the same as the speed of the kite, which is 10 m/s. However, the actual speed at which the string is being let out is the derivative of the length of the string with respect to time (dL/dt), which involves calculating it at the moment when the kite is 250 m away:
dL/dt = dL/dx ⋅ dx/dt
Here, dL/dx is the derivative of the length with respect to the horizontal distance (which is equal to x divided by L), and dx/dt is the horizontal speed of the kite (10 m/s). We can say:
dL/dt = (250 m / 291.55 m) ⋅ (10 m/s) = 8.57 m/s
Hence, the speed at which the string is being let out is approximately 8.57 m/s.