To find the velocity of the car, we need to integrate the acceleration function with respect to time, as the velocity is the integral of the acceleration function.

Given that the acceleration a(t) = 15cos(πt), we can integrate it to find the velocity v(t):
∫a(t) dt = ∫15cos(πt) dt

Using the standard integral of cosine function, we can calculate the integral:
v(t) = ∫15cos(πt) dt = (15/π)sin(πt) + C

With an initial speed of 51 mph, we can use this information to find the constant C. At t = 0 (initial time), the velocity should be 51 mph, so we have:
v(0) = (15/π)sin(π(0)) + C = 51
C = 51 - (15/π)sin(0) = 51

Therefore, the equation for the velocity v(t) is:
v(t) = (15/π)sin(πt) + 51

To find the driver's velocity after 40 minutes of driving, we substitute t = 2/3 (40 minutes = 2/3 hours) into the equation:
v(2/3) = (15/π)sin(π(2/3)) + 51

Calculating the sine function, we have:
v(2/3) = (15/π)sin(2π/3) + 51

Since sin(2π/3) = sin(π/3) = √3/2, we can plug in this value and calculate the velocity:
v(2/3) = (15/π)(√3/2) + 51
= (15√3)/(2π) + 51

Thus, the driver's velocity after 40 minutes of driving is (15√3)/(2π) + 51 mph.

Got a question with my answer?

Message Me

Cheapmath.com is completely free!

Asking question is free. Getting answers is free.

+99

Continue