If x2 + y2 = 100 and dy/dt = 9, find dx/dt when y = 8. (Enter your answers as a comma-separated list.)
To find dx/dt when y = 8, we need to differentiate the equation x^2 + y^2 = 100 implicitly with respect to t (time).
Differentiating both sides of the equation with respect to t, we get:
2x(dx/dt) + 2y(dy/dt) = 0
Now, we can substitute the given value of dy/dt = 9 and y = 8 into the equation:
2x(dx/dt) + 2(8)(9) = 0
Simplifying this expression:
2x(dx/dt) + 144 = 0
2x(dx/dt) = -144
Now, we can solve for dx/dt:
dx/dt = -144 / (2x)
However, we still need to find the value of x when y = 8.
Let's substitute y = 8 into the original equation:
x^2 + (8)^2 = 100
x^2 + 64 = 100
x^2 = 100 - 64
x^2 = 36
x = ± √36
x = ± 6
Now we have the two possible values for x, x = 6 and x = -6.
Plugging each value into the equation for dx/dt:
dx/dt for x = 6: dx/dt = -144 / (2(6)) = -144 / 12 = -12
dx/dt for x = -6: dx/dt = -144 / (2(-6)) = -144 / -12 = 12
Therefore, when y = 8, the values for dx/dt are -12 and 12.