at what rate is the angle shown changing at that instant? x=8 y=6 z=10
dx/dt=-15 dy/dt=10 dz/dt=-6
so i tried it by doing
sin(0)=y/z
(d0/dt)cos(0)=(dy/dt)/(dz/dt)
(d0/dt)(8/10)=(-10/6)
(d0/dt)=-25/12 rad/hour
did i miss it because i didn't do the quotent rule?
It appears so. y and z are functions of t, so you need the quotient rule.
Ty
To find the rate of change of the angle at a given instant, you can use the concept of related rates.
First, let's denote the angle as θ. In this case, sin(θ) = y/z. Since we are given that y = 6 and z = 10, we can substitute these values into the equation: sin(θ) = 6/10.
Taking the derivative of both sides of the equation with respect to time t, we have:
d/dt(sin(θ)) = d/dt(6/10)
Using the chain rule, we get:
cos(θ) * dθ/dt = (0 - 0) / 10
Since sin(0) = 0 and cos(0) = 1, the equation simplifies to:
dθ/dt = 0.
So, at that instant when x = 8, y = 6, and z = 10, the angle θ is not changing at all. The rate of change of the angle is zero.
Now let's analyze your working:
You wrote sin(0) = y/z, which is correct.
Next, you took the derivative of both sides to find (dθ/dt) * cos(0) = (dy/dt) / (dz/dt). But here, you made an error. The derivative of sin(θ) is not (dθ/dt), it is actually (dθ/dt) * cos(θ).
So, to find (dθ/dt), you have to rearrange the equation sin(θ) = y/z as cos(θ) * (dθ/dt) = (dy/dt) / (dz/dt).
Now, substituting the given values, we get:
cos(θ) * (dθ/dt) = (10/6)
The cos(θ) in this case is cos(0) (because sin(0) = y/z), which is equal to 1.
So, we have:
(dθ/dt) = (10/6)
Therefore, you made an error in the calculation. The correct value for (dθ/dt) is 10/6 rad/hour or approximately 1.67 rad/hour.
Note: The quotient rule is not required to solve this problem.