A $10$ kilogram object suspended from the end of a vertically hanging spring stretches the spring $9.8$ centimeters. At time $t = 0$, the resulting mass-spring system is disturbed from its rest state by the force $F(t) = 50\cos\!\left(8t\right)$. The force $F(t)$ is expressed in Newtons and is positive in the downward direction, and time is measured in seconds.

1. Determine the spring constant $k$.
   $k =$ Newtons / meter
   
   
2. Formulate the initial value problem for $y(t)$, where $y(t)$ is the displacement of the object from its equilibrium rest state, measured positive in the downward direction. (Give your answer in terms of $y, y^{\,\prime}, y^{\,\prime\prime}, t$.)
   
   Differential equation: help (equations)
   
   Initial conditions: $y(0) =$ and $y^{\,\prime}(0) =$ help (numbers)
   
   
3. Solve the initial value problem for $y(t)$.
   $y(t) =$ help (formulas)
   
   
4. Plot the solution and determine the maximum excursion from equilibrium made by the object on the time interval $0 \leq t < \infty$. If there is no such maximum, enter NONE.
   maximum excursion = meters help (numbers)