A rotating wheel requires 5 s to rotate 38 revolutions. Its angular velocity at end of time interval 5-s is 79 rad/s. What is the constant angular acceleration of the wheel? Show your work please

"Rotating wheel" is meant to indcate that the wheel is already rotating at the start. Denote the initial angular velocity by ω₀. Then the angular displacement θ at time t is

θ = ω₀t + 1/2 αt ²

while the angular velocity ω is

ω = ω₀ + αt

It takes 5 s for the wheel to rotate 38 times, or turn a total of (2π rad/rev) (38 rev) = 76π rad, as well as to reach an angular velocity of 79 rad/s, so that

76π rad = ω₀ (5 s) + 1/2 α (5 s)²

79 rad/s = ω₀ + α (5 s)

Solve the second equation for ω₀ and substitute into the first equation, then solve for α :

ω₀ = 79 rad/s - α (5 s)

==>   76π rad = (79 rad/s - α (5 s)) (5 s) + 1/2 α (5 s)²

==>   76π rad = (79 rad/s) (5 s) - 1/2 α (5 s)²

==>   1/2 α (5 s)² = (79 rad/s) (5 s) - 76π rad

==>   α = ((79 rad/s) (5 s) - 76π rad) / (1/2 (5 s)²)

==>   α ≈ 12.5 rad/s²

see earlier