Sinusoidal Function

[Chaim]

Michael’s weight fluctuates according to a sinusoidal function of time. Let x = 0 correspond to the beginning of the year (January, 1st). Because of his new year resolutions, Michael loses weight and reaches a minimum of 205 pounds at x = 60 days. Michael then stops exercising and dieting and his weight increases to a maximum of 250 pounds at x = 140 days.

If summer starts on the day x = 171 and ends on the day x = 265, how many days of summer will Michael’s weight be below 220 pounds (these are the days he fits into his bathing suit)?

Here is what I got:
Using the sinusoidal function Asin((2pi/B)(x-C))+D
I got 22.5sin((2pi/160)(x-100))+227.5


So where do I go after I go the equation?
ALSO, another curious question (if anyone got time or wants to answer it), how do you get rid of a sin in an equation.

Thanks!

 

[wjm11]

Michael’s weight fluctuates according to a sinusoidal function of time. Let x = 0 correspond to the beginning of the year (January, 1st). Because of his new year resolutions, Michael loses weight and reaches a minimum of 205 pounds at x = 60 days. Michael then stops exercising and dieting and his weight increases to a maximum of 250 pounds at x = 140 days.

If summer starts on the day x = 171 and ends on the day x = 265, how many days of summer will Michael’s weight be below 220 pounds (these are the days he fits into his bathing suit)?

Here is what I got:
Using the sinusoidal function Asin((2pi/B)(x-C))+D
I got 22.5sin((2pi/160)(x-100))+227.5


So where do I go after I go the equation?
ALSO, another curious question (if anyone got time or wants to answer it), how do you get rid of a sin in an equation.

I suggest you graph it and study the graph first. On a graphing calculator, set y1 = your equation, then set y2 = 220. Find the intersections in the required domain (x = 171 to x = 265). You might also find the TABLE function on your calculator to be of use.

To solve algebraically, set 220 = your equation. To "get rid of" the sine, use the arcsine (or inverse sine) function. Your calculator will only give you one solution. Study the graph and the periodicity of the function to determine other solutions. (Hint: you already know the period.)

Hope that helps.