# Thread: Another Related Rate Problem

1. ## Another Related Rate Problem

Here is the text of my problem:

A boy is flying a kite at a height of 150ft. If the kite moves gorizontally away from the boy at 20ft/s, how fast is the string being paid out when the kite is 250ft from him?

Given:
• y=150ft
• x=250ft
• $\displaystyle \frac{dx}{dt} = 20ft/s$
Find:
• $\displaystyle \frac{dz}{dt} when x = 250$
Work:
• by the theorem of pythagoras:
• $\displaystyle z = 50\sqrt{34}$
• $\displaystyle \displaystyle\frac{dz}{dt} = \frac{x*\frac{dx}{dt} + y*\frac{dy}{dt}}{z}$

• $\displaystyle \displaystyle\frac{dz}{dt} = \frac{(250*20)+ (150*0)}{50\sqrt{34}}$

• $\displaystyle \displaystyle\frac{dz}{dt}=\frac{5000}{50\sqrt{34} }= \frac{100}{\sqrt{34}}}$
The book goves an answer of 16ft/s. Can someone give me a hint of where I might have gone astray? Thanks!

2. Originally Posted by dbakeg00
Here is the text of my problem:

A boy is flying a kite at a height of 150ft. If the kite moves gorizontally away from the boy at 20ft/s, how fast is the string being paid out when the kite is 250ft from him?

Given:
• y=150ft
• x=250ft
• $\displaystyle \frac{dx}{dt} = 20ft/s$
Find:
• $\displaystyle \frac{dz}{dt} when x = 250$
Work:
• by the theorem of pythagoras:
• $\displaystyle z = 50\sqrt{34}$
• $\displaystyle \displaystyle\frac{dz}{dt} = \frac{x*\frac{dx}{dt} + y*\frac{dy}{dt}}{z}$

• $\displaystyle \displaystyle\frac{dz}{dt} = \frac{(250*20)+ (150*0)}{50\sqrt{34}}$

• $\displaystyle \displaystyle\frac{dz}{dt}=\frac{5000}{50\sqrt{34} }= \frac{100}{\sqrt{34}}}$
The book goves an answer of 16ft/s. Can someone give me a hint of where I might have gone astray? Thanks!
The distance from the person is the hypotenuse not the adjacent side.

3. ok, I see what you mean.

That gives me this triangle:
x=200
y=150
z=250

which now gives me:
$\displaystyle \displaystyle\frac{dz}{dt}=\frac{(x*\frac{dx}{dt}) +(y*\frac{dy}{dt})}{z}$

$\displaystyle \displaystyle\frac{dz}{dt}=\frac{(200*20)+(150*0)} {250}$

$\displaystyle \displaystyle\frac{dz}{dt}=\frac{4000}{250}=16ft/s$

Thanks for the help, appreciate it very much. These word problems are sometimes difficult for me to visualize if I'm not careful.

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### a boy is flying a kite at a height of 50m if the kite moves horizontally away from the boy at the rate of 6m per sec how fast is the string being paid out when the kite is 80m from him

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