The height of an object moving up and down is described by y=72+40cos18t (in degree mode). To at least three decimal places, confirm that the average speed of the object is 2/pi times its greatest speed.

Answer 1

The period of cos 18t is 360°/18 = 20°
so the change of t from min to max is 10° or π/18

the min of 72+40cos18t is 72 - 40 = 32
and the max of 72+40cos18t is 72+40 = 112

avg rate of change = (112-32)/(π/18) = 1440/π

greatest speed is at the inflection point of y=72+40cos18t
y' = -720sin 18t <<<<----- speed
y'' = - 12960 cos 18t = 0 for point of inflection

cos 18t = 0
18t = π/2 , 3π/2 , ....
t = π/36 , π/12, .....

speed is -720sin 18t
when t = π/36, speed = -720sin π/2 = -720
when=n t = π/12, speed = -720sin 3π/2 = + 720

greatest speed/avg speed = (1440/π) / 720 = 2/π

yeahhhh!
I will let you calculate that value to 3 decimals, I did not need a calculator for the above