Need help solving this word problem.. Implicit Differentiation?

**Manizzle** · October 22nd 2008, 04:10 PM

I believe this problem needs to be solved using implicit differentiation, but i am not sure how to set the problem up.. any help is greatly appreciated.

People standing on a cliff watch a sailboat move away from the base of the cliff. The boat is traveling at 3 m/s and the cliff is 150 meters tall. Not taking into account the earth’s curvature, how fast is the distance between the people and the boat increasing when the boat is 200 meters from the base of the cliff?

**Shyam** · October 22nd 2008, 04:58 PM

See the diagram attached,
In the right-angled Triangle,

$x^2+y^2=z^2 \;\;.............eqn(1)$

$(150)^2+(200)^2=z^2$

z = 250

Now, differentiate eqn(1), w.r.t. "t", (time),

$2x \frac{dx}{dt}+2y \frac{dy}{dt}=2z \frac{dz}{dt}$

$x \frac{dx}{dt}+y \frac{dy}{dt}=z \frac{dz}{dt}\;\;.......eqn(2)$

also, boat is moving away from base of cliff at the rate 3 m/s, so,

$\frac{dy}{dt}=3$

also, $\frac{dx}{dt}=0$ because the height of cliff is fixed, it does not change.

Put these values in eqn(2),

$<br /> (150) (0)+(200) (3)=(250) \frac{dz}{dt}$

$\frac{dz}{dt}= \frac{600}{250}= \frac{12}{5}$

so, the distance between the people (at the top of cliff) and the boat is increasing at the rate of $\frac{12}{5}$ m/s.

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