Answer:
dy/dt = -6/5
Step-by-step explanation:
We are given that x = x(t) and y = y(t) are both functions of t and x² + y² = 34.
Differnatiate both sides of x² + y² = 34 with respect to t.
The equation becomes:
[tex] \sf \dfrac{d}{dt}(x^2 + y^2) = 0 [/tex]
[tex] \sf 2x \bigg ( \dfrac{dx}{dt} \bigg) + 2y \bigg (\dfrac{dy}{dt} \bigg )= 0 [/tex]
simplifying the equation,
[tex] \sf x \bigg (\dfrac{dx}{dt} \bigg )+ y \bigg (\dfrac{dy}{dt} \bigg ) = 0 [/tex]
We need to find the value of dy/dy, when x = 3 and y = 5. Also we are given that dx/dt = 2.
To solve for dy/dy, substitute the values of x, y and dx/dt in above equation :
[tex] \sf 3 (2) + 5 \bigg( \dfrac{dy}{dt}\bigg ) = 0 [/tex]
[tex] \sf 6 + 5 \bigg( \dfrac{dy}{dt}\bigg ) = 0 [/tex]
[tex] \sf 5 \bigg( \dfrac{dy}{dt} \bigg ) = - 6 [/tex]
[tex] \sf \dfrac{dy}{dt} = \dfrac{- 6}{5} [/tex]
Therefore, when x = 3 and y = 5, the value of dy/dt is -6/5.
