Firstly, we need to find the components of the velocity vector. The velocity components can be computed by taking the derivative of each component of the position function with respect to time 't'. For this case, the 'i' and 'j' components of the position function are 3t⁴ + 5t² and 6t³ + 7t respectively.
Now, let's first differentiate the 'i' component of the position function with respect to time:
di/dt = d/dt(3t⁴ + 5t²)
Using the power rule of differentiation gives us:
di/dt = 4 * 3 * t³ + 2 * 5 * t.
Substituting t = 5 s into this equation we get:
di/dt = 1550 m/s.
Similarly, let's find the derivative of the 'j' component with respect to time:
dj/dt = d/dt(6t³ + 7t)
Once again, using the power rule of differentiation we get:
dj/dt = 3 * 6 * t² + 7
Substituting t = 5 s into this equation, we get:
dj/dt = 457 m/s.
So the velocity vector at time t= 5 s is (1550 i + 457 j) m/s.
The magnitude of velocity is given by the square root of the sum of the squares of its components:
magnitude_velocity = sqrt[(di/dt)² + (dj/dt)²]
Substituting the values we have calculated into this equation:
magnitude_velocity = sqrt[(1550)² + (457)²] = 1615.97 m/s (approximately).
Therefore, the magnitude of the velocity of the object at t = 5s is around 1615.97 m/s.