Lauren is playing with a yo-yo and keeping track of its height over time. She has plotted some key points on the graph at right. The $y$-values are in inches and the $x$-values are in seconds. Complete the following problems based on the graph.  

1. How much time does it take to complete one cycle?

   Hint (a):

   Use the $x$-values to determine this quantity.

2. How long is the string?

   Hint (b):

   This is the difference in the maximum and minimum heights.

3. Write an equation that models the height of the yo-yo over time.

   Hint (c):

   Identify:
   1. Amplitude
   2. Vertical Shift
   3. Horizontal Shift
   4. $b$ (frequency)
   
   Note: Part (a) is the period and $b=\frac{2\pi}{p}$
   Test your answer in your graphing calculator.

4. Determine the first two times when the yo-yo is exactly $30$ inches above the ground.

   Hint (d):

   Use a graphing calculator to determine where the function crosses $y = 30$.
   Note: The first two times may have occurred before the curve in the given sketch.