If x^2 + y^2 = 100 and dy/dt = 6, find dx/dt when y = 8. (Enter your answers as a comma-separated list.)
wqe
A street light is mounted at the top of a 15-ft-tall pole. A man 6 ft tall walks away from the pole with a speed of 7 ft/s along a straight path. How fast is the tip of his shadow moving when he is 40 ft from the pole?
It is wrong
To find dx/dt when y = 8, we can use implicit differentiation.
First, let's differentiate both sides of the equation x^2 + y^2 = 100 with respect to t (the independent variable).
d/dt(x^2 + y^2) = d/dt(100)
Using the chain rule, we have:
2x * dx/dt + 2y * dy/dt = 0
Since dy/dt is given as 6, we can substitute that value into the equation:
2x * dx/dt + 2y * 6 = 0
Simplifying the equation, we get:
2x * dx/dt + 12y = 0
Now, we can substitute y = 8 into the equation, since we want to find dx/dt when y = 8:
2x * dx/dt + 12 * 8 = 0
2x * dx/dt + 96 = 0
Subtracting 96 from both sides, we have:
2x * dx/dt = -96
Finally, we divide both sides by 2x to solve for dx/dt:
(dx/dt) = -48/x
So, when y = 8, dx/dt is equal to -48/x.
Please note that since the value of x is not given in the question, we cannot provide a specific numeric value for dx/dt.
take the derivative with respect to t
2x dx/dt+2ydy/dt=0
dx/dt= -y/x dy/dt
so when y=8, find x, or x=+-sqrt(100-64)
x=+-6
dx/dt=put the values in and compute. Yes, you have two solutions, the x^2+y^2 is not a function, it does not obey simple rules. Think about it.
Your computer tutor may not be that smart.