A girl flies a kite at a height of 200 ft, the wind carrying the kite horizontally away from her at a rate of 4 ft/sec. How fast must she let out the string when the kite is 422 ft away from her?
Answer = ft/sec.
To find the speed at which she must let out the string, we can use the concept of related rates.
Let's start by assigning variables to the given values:
- The height of the kite above the ground: let's call it "h" (h = 200 ft).
- The distance between the girl and the kite along the ground: let's call it "x" (we need to find the value of x when the kite is 422 ft away).
We are given that the wind is carrying the kite horizontally away from the girl at a rate of 4 ft/sec. This means that the rate of change of x with respect to time (dx/dt) is 4 ft/sec.
We need to find the rate at which the girl must let out the string, which is the rate of change of h with respect to time (dh/dt) when x = 422 ft.
To find the relationship between h, x, and the rate at which the string is being let out (dh/dt), we can use the Pythagorean theorem:
x^2 + h^2 = (distance between the girl and the kite)^2.
Substituting the given values, we have:
422^2 + 200^2 = h^2.
Next, we can differentiate this equation with respect to time (t):
(2x)(dx/dt) + 2h(dh/dt) = 0.
Plugging in the given values, we have:
(2 * 422)(4 ft/sec) + 2(200)(dh/dt) = 0.
844(4) + 400(dh/dt) = 0.
Now we can solve for dh/dt, which represents the rate at which the string is being let out:
dh/dt = [-(844 * 4)] / 400.
Calculating this expression, we find:
dh/dt = -33.76 ft/sec.
However, the negative sign indicates that the string is being pulled in, rather than let out. So, the magnitude of the value is:
|dh/dt| = 33.76 ft/sec.
Therefore, the girl must let out the string at a speed of 33.76 ft/sec when the kite is 422 ft away from her.