Amass is hung from a spring and set in motion so that it oscillates continually up and down. the velocity v of the weight at time t is given by the equation y=−5 cos(4πt) with v measured in feet per second and t measured in seconds. determine the maximum velocity of the mass and the amount of time it takes for the mass to move from its lowest position to its highest position.

To solve the problem it is necessary to identify the equation in the manner given above.

This equation corresponds to the displacement of a body under the principle of simple harmonic movement.

Where,

PART A) Our equation corresponds to

Therefore the value of omega is equivalent to that of

From the definition we know that the period as a function of angular velocity is equivalent to

This same point is the equivalent of the maximum point of the speed that the body can reach, since the internal expression of the Is equivalent to . So the maximum speed that the body can reach is,

Therefore the maximum felocity will be 5ft / s

PART B) The period of graph is the time taken to reach from one maximum point to next point maximum point, then

we have been given a quadratic function and we need to restrict the domain such that it becomes a one to one function.

we know that vertex of this quadratic function occurs at (5,2).

further, we know that range of this function is .

if we restrict the domain of this function to either or , it will become one to one function.

let us know find its inverse.

upon interchanging x and y, we get:

let us now solve this function for y.

hence, the inverse function would be if we restrict the domain of original function to and the inverse function would be if we restrict the domain to .