For time 0 ≤ t ≤ 8, people arrive at a venue for an outdoor concert at a rate modeled by the function A defined by A(t) = 0.3sin(1.91)+ 0.3cos(0.61) + 1.3. For time 0 ≤ t ≤ 1, no one leaves the venue, and for time 1 ≤ t ≤ 8, people leave the venue at a rate modeled by the function L defined by L(t) = 0.2cos(1.91) + 0.2sint + 0.8. Both A(t) and L(t) are measured in hundreds of people per hour, and t is measured in hours. The number of people at the venue, in hundreds, at time 1 hours is given by P(t).
(a) At time t = 2, there are 6 hundred people at the venue. Write an equation for the locally linear approximation of P at t = 2, and use it to approximate the number of people at the venue at time t = 12.5.