Q: If y3=16x2, find dx/dt when x=4 and dy/dt=-1
Answer:
First thing is to take the derivative of the equation implicitly to get:
(3y2)dy/dt = (32x)dx/dt
The goal is to solve for dx/dt… So we need to know y, x, and dy/dt. We know x and dy/dt (given in the problem). Use the original equation to solve for y when x is 4:
y3=16(4)2
which gives:
y = cuberoot(256)
So, now we know all that we need to know to solve for dx/dt:
(3y2)dy/dt = (32x)dx/dt
Plug in each variable:
(3(cuberoot256)2)(-1) = (32*4)dx/dt
Simplify:
-3*32*cuberoot(2) = (32*4)dx/dt
(-3/4)cuberoot(2) = dx/dt