The displacement x of a particle at time t is given by x = 5sin(4t) + 12cos(4t) - HSC - SSCE Mathematics Extension 1 - Question 7 - 2016 - Paper 1

To find the maximum velocity, we need to express the velocity in terms of a single trigonometric function. This can be achieved using the R-Method (Amplitude-Phase form):

Let:

$R = ext{sqrt}(a^2 + b^2)$

where:

a = 20 and b = -48.

Calculating R:

$R = ext{sqrt}(20^2 + (-48)^2) = ext{sqrt}(400 + 2304) = ext{sqrt}(2704) = 52$

Thus, we can express the velocity as:

$v(t) = Rsin(4t + heta)$

where $\theta$ is the phase shift. The maximum value of sin is 1, therefore:

$v_{max} = R = 52.$