A kite is being flown at a height of 300 ft, and the wind carries the kite horizontally away at a rate of 25ft/sec. How fast must the string be let out when the kite is 500ft away?
25 ft/sec
To find the speed at which the string must be let out, we need to use the concept of related rates. Let's break down the information given:
- The kite is flying at a height of 300 ft.
- The wind carries the kite horizontally away at a rate of 25 ft/sec.
We want to determine how fast the string must be let out when the kite is 500 ft away. To do this, we can use the Pythagorean Theorem to relate the height and horizontal distance of the kite.
Let's denote the height of the kite as "h" and the horizontal distance as "x." We know that the height is 300 ft and the horizontal distance is changing with respect to time at a rate of 25 ft/sec.
Using the Pythagorean Theorem, we have:
x^2 + h^2 = d^2
where d is the distance between the kite and the person on the ground. We know that d is 500 ft.
Now, let's differentiate both sides of this equation with respect to time:
d/dt (x^2 + h^2) = d/dt (d^2)
2x(dx/dt) + 2h(dh/dt) = 0
Since the height of the kite remains constant at 300 ft, we have dh/dt = 0.
Simplifying the equation further, we have:
2x(dx/dt) = 0
Now, we can solve for dx/dt, which represents the rate at which the horizontal distance is changing:
dx/dt = 0 / (2x)
= 0
Therefore, the rate at which the string must be let out when the kite is 500 ft away is 0 ft/sec. This means that the string does not need to be let out any further when the kite is at a distance of 500 ft.