particle moves along a hyperbola xy = 12. As it reaches the point (4, 3), 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?
Question
particle moves along a hyperbola xy = 12. As it reaches the point (4, 3), 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?
Solution
This problem involves implicit differentiation, a concept in calculus. Here are the steps to solve it:
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Given the equation of the hyperbola, xy = 12, differentiate both sides with respect to time (t). This gives us x(dy/dt) + y(dx/dt) = 0, where dy/dt and dx/dt are the rates of change of y and x respectively.
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We are given that dy/dt = -3 cm/s (negative because y is decreasing) and we need to find dx/dt. We also know that at the point (4,3), x = 4 and y = 3.
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Substitute these values into the differentiated equation: 4(-3) + 3(dx/dt) = 0.
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Simplify this to get -12 + 3(dx/dt) = 0.
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Solve for dx/dt to get dx/dt = 12/3 = 4 cm/s.
So, the x-coordinate is changing at a rate of 4 cm/s at that instant.
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