For what values of t is the speed of the particle increasing?

The position of the particle is given by the function s(t) = 2t³ – 24t² + 90t + 7.

To determine where the speed of the particle is increasing, we first need to find the velocity of the particle by taking the derivative of the position function:

v(t) = s'(t) = rac{d}{dt}(2t³ – 24t² + 90t + 7) = 6t² – 48t + 90.

The speed of the particle is increasing when the velocity is increasing, which occurs when the acceleration is positive. To find the acceleration, we take the derivative of the velocity function:

a(t) = v'(t) = rac{d}{dt}(6t² – 48t + 90) = 12t – 48.

Setting the acceleration greater than zero gives us:

12t – 48 > 0

Simplifying this inequality:

12t > 48

t > 4

Therefore, the speed of the particle is increasing for t > 4.