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 need to find the critical points by setting the derivative equal to zero:

$0 = 20\text{cos}(4t) - 48\text{sin}(4t).$

Rearranging gives us:

$\frac{20}{48} = \frac{\text{sin}(4t)}{\text{cos}(4t)} = \tan(4t).$

Letting this be equal to a constant k, we can find the angle:

$4t = \tan^{-1}\left(\frac{20}{48}\right)$

Calculating this gives us the maximum value of the sine and cosine functions since they range from -1 to 1. The maximum result can be achieved by calculating the amplitude:

Using the Pythagorean identity, the maximum velocity can be approximated by:

$\sqrt{(20)^2 + (48)^2} = \sqrt{400 + 2304} = \sqrt{2704} = 52.$