A girl is flying a kite at a height of 320 ft. If the wind blows the kite horizontally at a rate of 15 feet per second away from the girl, at what rate is the girl releasing the string when the length of the string is 400 ft?

[kite] <---15 ft/sec
. .
. .
. .
. .
. .
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320 ft .
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. .
._____________.[girl]

So I listed my given data:
a=320 ft
b=240 ft
c=400 ft

And I know I have to find dc/dt when c=400 ft - using the Pythagorean theorem. I differentiated it but now I realize that I don't have da/dt and db/dt...what do I do?

I can see from your data that you labeled your triangle with a, b, and c

a being the vertical (fixed at 320) , b the horizontal and c the hypotenuse (the string)

so c^2 = b^2 + 320^2
you also found correctly that if c= 400, b = 240
differentiate ...
2c dc/dt = 2b db/dt + 0
dc/dt = b db/dt/c
= 240(15)/400 = 9

check my arithmetic

Ohh thank you-that is what I had overlooked: the fact that the derivative of a constant is = to 0 and that is why I did not need da/dt. Thank You I understand where I went wrong now.

I was going to mention, that since a is a constant, da/dt = 0 as seen above.

To solve this problem, we can use the Pythagorean theorem to relate the variables a, b, and c:

c^2 = a^2 + b^2

First, differentiate both sides of the equation with respect to time t:

2c * (dc/dt) = 2a * (da/dt) + 2b * (db/dt)

Since we are interested in finding (dc/dt) when c = 400 ft, we need to find (da/dt) and (db/dt) when c = 400 ft.

Let's find (da/dt) first. We can use the given information that the girl is flying the kite at a height of 320 ft. Since a is the height and it is not changing, da/dt = 0.

Next, let's find (db/dt). We are given that the wind blows the kite horizontally at a rate of 15 ft/sec, which means the rate at which b is changing is (db/dt) = -15 ft/sec (negative because b is decreasing).

Now, plug in the values we found into the equation:

2(400) * (dc/dt) = 2(320) * (0) + 2(-15) * (-15)

Simplifying the equation:

800 * (dc/dt) = 450

Dividing both sides by 800:

(dc/dt) = 450/800

(dc/dt) = 0.5625 ft/sec

Therefore, when the length of the string is 400 ft, the girl is releasing the string at a rate of 0.5625 ft/sec.