The displacement x of a particle at time t is given by $$x = 5 ext{sin}(4t) + 12 ext{cos}(4t).$$ What is the maximum velocity of the particle? - HSC - SSCE Mathematics Extension 1 - Question 7 - 2016 - Paper 1

To find the maximum velocity, we can use the formula for the amplitude of a harmonic function. The maximum value of the function:

$v(t) = 20 ext{cos}(4t) - 48 ext{sin}(4t)$

can be determined using the formula:

$|A| = ext{√}(a^2 + b^2)$

where, in this case, a = 20 and b = -48.

Thus:

$|A| = ext{√}(20^2 + (-48)^2) = ext{√}(400 + 2304) = ext{√}(2704) = 52.$

Therefore, the maximum velocity of the particle is 52.