Question:
If x^2 + y^2 = 100 and dy/dt = 6, find dx/dt when y = 8.
(Enter your answers as a comma-separated list.)
Solution:
Givens & Unknowns:
x = ?
y = 8
dx/dt = ?
dy/dt = 6
Take the derivative explicitly to get:
2x dx + 2y dy = 0
Divide by 2 to simplify:
x dx + y dy = 0
Divide by dt:
x dx/dt + y dy/dt = 0
Solve for dx/dt:
dx/dt = -(y/x) dy/dt
Find x:
y = 8
x^2 + y^2 = 100
x = ±SQRT(100 - y^2)
x = ±SQRT(100 - 8^2)
x = ±SQRT(36)
x = ±6
Substitute:
dx/dt = -(y/x) dy/dt
for x = 6:
dx/dt = -(8/6)*6
= -8
For x = -6:
dx/dt = -(8/-6)*6
= 8
THEREFORE, dx/dt = 8, -8
