if y^3=9x^2, determine dx/dt when x=3 and dy/dx=3related rates

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You should differentiate the function `y^3 = 9x^2`  with respect to t such that:

`y = root(3)(9x^2) => y = (9x^2)^(1/3)`

`(dy)/(dx) = (1/3)(9x^2)^(1/3 - 1)*(18x)*(dx)/(dt)`

`(dy)/(dx) = (1/3)(9x^2)^(-2/3)*(18x)*(dx)/(dt)`

The problem provides the information that `(dy)/(dx) = 3`  and `x = 3` , hence, you may evaluate `(dx)/(dt)`  using the given values such that:

`3 = (1/3)(81)^(-2/3)*(18*3)*(dx)/(dt) => 3 = 18/(root(3)(3^6))*(dx)/(dt)`

`1 = 6/9*(dx)/(dt) => (dx)/(dt) = 9/6 => (dx)/(dt) = 3/2`

Hence, evaluating `(dx)/(dt)`  under the given conditions yields `(dx)/(dt) = 3/2.`

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