#### Answer

The velocity of the particle after $t$ seconds is $$\frac{5}{2}\pi\cos(10\pi t) (cm/s)$$

#### Work Step by Step

$$s(t)=10+\frac{1}{4}\sin(10\pi t)$$
From physics, the velocity of the particle is the rate of change of its displacement.
That means, to get the formula of the velocity, we need to get the derivative of the displacement formula.
In other words, $$v(t)=s'(t)=[10+\frac{1}{4}\sin(10\pi t)]'$$ $$v(t)=\frac{1}{4}\frac{d(\sin(10\pi t))}{dt}$$
Apply the Chain Rule: $$v'(t)=\frac{1}{4}\frac{d(\sin(10\pi t))}{d(10\pi t)}\frac{10\pi dt}{dt}$$ $$v'(t)=\frac{1}{4}\cos(10\pi t)\times10\pi\times1$$ $$v'(t)=\frac{5}{2}\pi\cos(10\pi t)$$
In conclusion, the velocity of the particle after $t$ seconds is $$\frac{5}{2}\pi\cos(10\pi t) (cm/s)$$