I need help setting up this equation and what percentages are percentages of? Thanks

Question

Flowers tend to open in the daylight and close in the dark. The percent P of flowers open during a 24-hour period can be modeled by P(x) = a sin (bx) + d, where x is hours past 6 A.M. (a) Suppose that for P the maximum is 20%, and the minimum is 10%, and the period is 24 hours. Determin P(x). Any insight to how to model this equation?

Answer 1

Hey Nathan.

 

The first thing you should look at is b. Your question states that the period is 24 hours. Your variable b is directly related to the period by:

 

period = 2pi/b

 

24 = 2pi/b

 

b = pi/12

 

Next up is a. This represents the amplitude of your sine wave. Since you oscillate between 20% and 10%, it means your amplitude is

 

(20 - 10)/2 = 5.

 

If you go up 5 units and down 5 units, it means you span 10 units in total on your wave, which is just what your curve does.

 

a = 5

 

Finally you need to look at d. This represents your vertical displacement. If your vertical displacement is 0, your wave will have a minimum value of -5 and a max value of +5 based on your amplitude determined above. It will start at 0, go up to 5, and down to -5.

 

However, your problem states that the minimum value is 10 and the max value is 20. That means that your curve has to be shifted up by 15 units to account for this.

 

-5 + 15 = 10

5 + 15 = 20

 

d = 15

 

Your percent function therefore is:

 

P(x) = 5*sin(pi/12*x) + 15

 

Hope this helps.