After 14.6 s, a spinning roulette wheel has slowed down to an angular velocity of 1.22 rad/s. During this time, the wheel has an angular acceleration of -4.51 rad/s2. Determine the angular displacement of the wheel.

I did
w0= wf + (4.51)(14.6) = 1.22+45.1

46.32(14.6) - (1/2)(4.51)(14.6)(14.6) = 159

Did I do this correctly?

To determine the angular displacement of the wheel, you can use the equations of motion for rotational motion.

First, let's clarify the given values:
Initial angular velocity (ω0) = 0 rad/s (since it starts from rest)
Final angular velocity (ωf) = 1.22 rad/s
Angular acceleration (α) = -4.51 rad/s^2
Time (t) = 14.6 s

To find the angular displacement (θ), we can use the equation:

θ = ω0t + (1/2)αt^2

Substituting the given values, we have:

θ = (0)(14.6) + (1/2)(-4.51)(14.6)^2
θ = (1/2)(-4.51)(213.16)
θ = -481.06 rad

From your calculations, it seems you made a mistake while calculating ω0t. The correct calculation should be:
ω0t = 0 + (-4.51)(14.6) = -65.85 rad/s

Therefore, the correct angular displacement of the wheel is -481.06 rad. Keep in mind that the negative sign indicates that the wheel has rotated in the clockwise direction.

ω=ωₒ - ε•t,

ωₒ= ω+ ε•t =1.22+4.51•14.6 =67.1 rad/s.
φ = ωₒ•t - ε•t²/2 =67.1•14.6 – 4.51(14.6)²/2 =1468 rad