The movement of the body is described by the equation x = -0.25t ^ 4 + 3t ^ 2 + 72. Find the maximum body speed.

Given:

x = -0.25 * t ^ 4 + 3 * t ^ 2 + 72 – the equation of body motion.

It is required to determine the maximum body speed v (m / s).

At the initial moment of time t = 0, the body was at the point x0 = 72.

At time t, the body is at the point x = -0.25 * t ^ 4 + 3 * t ^ 2 + 72.

Then the speed of the body will be equal to:

v = (x – x0) / t = (-0.25 * t ^ 4 + 3 * t ^ 2 + 72 – 72) / t =

= -0.25 * t ^ 4 + 3 * t ^ 2 / t = -0.25 * t ^ 3 + 3 * t.

By analyzing the function, we find that the maximum speed value will be at t = 2 seconds:

v = -0.25 * 2 ^ 3 + 3 * 2 = -0.25 * 8 + 6 = -2 + 6 = 4 m / s.

Answer: The maximum speed will be 4 m / s.