Maria started observing Elasticmans height at midnight At 3

Maria started observing Elasticman\'s height at midnight. At 3 AM, he was at his shortest: only 3 feet tall. At 9 AM, he was at his tallest: 11 feet tall. Elasticman\'s height is a sinusoidal function of time. In the 24 hours after Maria began observing Elasticman, how much of the time will Elasticman be less than 4 feet tall? ()

Solution

general form of sine equation: y=A sin B(x-C) + D

A= (max y - min y)/2= (11-3)/2 = 4
B= 2/T; T=(time between max x and min x) * 2= (9-3) * 2 = 12
therefore B= 2/12= /6
C= -(time to max x - T/4) = -(9 - 3) = -6
D= (max y + min y)/2 = (11 + 3)/2 = 7

y = 4 sin (/6(x+6)) + 7
In graph (x is hours since midnight, y is height of elastic man)

now we gotta find the time for which elasticman is below 4 feet tall.

y=4, 4 sin (/6(x+6)) + 7 = 4
4 sin (/6(x+6)) = -3 [take away 7 from both sides]
sin (/6(x+6)) = -3/4 [divide both sides by 4]
/6(x+6) = arcsin(-3/4) = -0.848 radians [take the inverse of sine on both sides]
general equation: /6(x+6) = n + (-1)^n * -0.848
*(x + 6) = 6n + (-1)^n *-5.088
x + 6 = 6n + (-1)^n * -1.62
x = 6n + ( (-1)^n * -1.62 ) - 6

substituting values of n= 1 and 2, this gives values of x, and therefore the hours elasticman\'s height is equal to 4 feet: 1.62 and 4.38

now we find the difference between these two to give the time in hours he is below 4 feet: 4.38 - 1.62 = 2.76 hours.This is over one 12 hour period so the time over 24 hours is 2.76 * 2 = 5.52 hours

elasticman\'s height is below 4 feet for 5.52 hours out of a 24 hour period