Question

Assume that x and y are both differentiable functions of t and find the required values of dy/dt and dx/dt. x2 + y2 = 25 (a) Find dy/dt, given x = 3, y = 4, and dx/dt = 5. dy/dt = -3/4 Incorrect: Your answer is incorrect. (b) Find dx/dt, given x = 4, y = 3, and dy/dt = –5.

Answer 1

(a)
`(dy)/(dt)=-3(3)/(4)`

(b)
`(dx)/(dt)=3(3)/(4)`

Explanation:

`x^(2) +y^(2)=25`

Take
`(d)/(dt)` of of each term.

`(d)/(dt)(x^(2))+(d)/(dt)(y^(2))=(d)/(dt)(25)\\\\((d)/(dx)(x^(2))*(dx)/(dt)) +((d)/(dy)(y^(2))*(dy)/(dt))=(d)/(dt)(25)\\\\2x(dx)/(dt) +2y(dy)/(dt) = 0\\\\`

For Question a

`2y(dy)/(dt)=-2x(dx)/(dt)\\\\(dy)/(dt)=(-2x(dx)/(dt))/(2y) \\\\(dy)/(dt)=-(x)/(y)(dx)/(dt)`

Given that x = 3, y = 4, and dx/dt = 5.

`(dy)/(dt)=-(3)/(4)*5=-(15)/(4)\\ \\(dy)/(dt)=-3(3)/(4)`

For Question b

`2x(dx)/(dt)=-2y(dy)/(dt)\\\\(dx)/(dt)=(-2y(dy)/(dt))/(2x) \\\\(dx)/(dt)=-(y)/(x)(dy)/(dt)`

Given that x = 4, y = 3, and dx/dt = -5.

`(dx)/(dt)=-(3)/(4)*-5=(15)/(4)\\ \\(dx)/(dt)=3(3)/(4)`