1. a ladybug lands on the end of a pinwheel that is spinning at a steady speed. the equation y = –7 sin (0.4πx) + 11 gives the height of the ladybug y, in centimeters, above the ground x seconds after landing. what is the diameter of the pinwheel

That question isn't nearly as hard as they want it to be.  The "a" in the sin function is the altitude of the function...how high it goes up and how low it goes down.  The negative in front of the 7 just means that it starts going down before it goes up (a classic sine wave goes up and then down; the negative just reverses that).  From the wave's midpoint on the y axis, which is 11 (hence, the +11 in the equation), the wave goes up 7 (to a y value of 18) and then goes down 7 (to a y value of 4).  Because it goes up 7 and then down 7, the diameter of the pinwheel is 14. The sin wave mimics the round and round motion of the pinwheel.  You could have been able to tell that it was 14 just by looking at what the altitude was.  The altitude is always the number the wave goes up and down from its midpoint. 7 up and 7 down makes 14 altogether.

The answer is 109 times per second