kathrynconley9
Answered

assume x=x(t) and y=y(t). Find dx/dt if x^2(y-6) = 12y + 3 and dy/dt = 2 when x = 5 and y = 12

Verified Answer :

LammettHash

\(x^2(y-6)=12y+3\)

Differentiate:

\(2x\dfrac{\mathrm dx}{\mathrm dt}(y-6)+x^2\dfrac{\mathrm dy}{\mathrm dt}=12\dfrac{\mathrm dy}{\mathrm dt}\)

Plug in everything you know: \(\dfrac{\mathrm dy}{\mathrm dt}=2\), \(x=5\), and \(y=12\)

\(2(5)\dfrac{\mathrm dx}{\mathrm dt}(12-6)+5^2(2)=12(2)\implies\dfrac{\mathrm dx}{\mathrm dt}=-\dfrac{13}{30}\)

The value of dx/dt at given conditions is -13/30.

How is dx/dt calculated?

Given Equation:

\(x^{2} (y-6)=12y+3\)

With the help of chain rule, we will differentiate the above eqn.

Differentiation of eqn w.r.t to t:

\(2x\dfrac{dx}{dt}(y-6)+x^{2}\dfrac{dy}{dt}=12\dfrac{dy}{dt}\)      {Since x and y are functions of t}

Given:

x=5, y=12, dy/dt=2  ,put values in above eqn.

\(2(5)\dfrac{dx}{dt}(12-6)+5^{2}(2) =12(2)\\\\ 60\dfrac{dx}{dt}=-26\\\\\dfrac{dx}{dt}=\dfrac{-13}{30}\)

Therefore value of dx/dt is -13/30.

To know more about differentiation: https://brainly.in/question/1238773

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