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Problem 12 Easy Difficulty

A particle is moving along a hyperbola $ xy = 8. $ As it reaches the point (4, 2), the $ y- $ coordinate is decreasing at a rate of $ 3 cm/s. $ How fast is the $ x- $ coordinate of the point changing at that instant?

Answer

The $x$ -coordinate is Increasing at the rate of 6 $\mathrm{cm} / \mathrm{s}$

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Video Transcript

until that a particle is moving along. A hyper bola X Y equals eight. We're told as it reaches the 0.42 Yeah. Mhm. The Y coordinate is decreasing at a rate of three centimeters per second. So the white teeth is negative. 3, 70 years ago, second were asked how fast the X coordinate of the point is changing at that. In other words, we want to find DX DT Now, To do this, I'm gonna differentiate our equation with respect to using implicit differentiation season the product rule This is DX DT times y plus x times dy DT and the right hand side is zero. I'm gonna solve this equation for the X t t. So the x p t equals negative X over y times divide et plugging in our values But we have the exes four wise to B Y t is negative three. Yeah, this is positive Oh six. 15 and the units are in centimeters per second. Therefore, it follows that the X coordinate is increasing since the sign is positive at a rate, uh, six centimeters per second. White is the most Chinese heroin