SOLUTION: The average temperature of a fictious town is given by the equation: f(x) = 20 sin (π/13x) + 2 where x represents the number of weeks since beggining of year. During the course of

Algebra ->  Trigonometry-basics -> SOLUTION: The average temperature of a fictious town is given by the equation: f(x) = 20 sin (π/13x) + 2 where x represents the number of weeks since beggining of year. During the course of      Log On


   

Question 1204408: The average temperature of a fictious town is given by the equation: f(x) = 20 sin (π/13x) + 2 where x represents the number of weeks since beggining of year. During the course of the year for how many weeks was the average temperature higher than 15° celcius
Found 2 solutions by Edwin McCravy, math_tutor2020:
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!


20sin%28expr%28pi%2F13%29x%29+%2B+2+=+15

20sin%28expr%28pi%2F13%29x%29+=+13

sin%28expr%28pi%2F13%29x%29+=+13%2F20

0.2416609734x+=+0.7075844367%2B2pi%2An or 0.2416609734x+=+2.434008217%2B2pi%2An
x+=+0.7075844367%2F0.2416609734%2B26n or x+=+2.43400817%2F0.2416609734%2B26n

x+=+2.928004579%2B26n or x+=+10.07199542%2B26n

It starts being above 15o on week 2.9 from the first equation when n=0 
and stops being above 15o on week 10.1 from the second equation when n=0.

It starts being above 15o on week 28.9 from the first equation when n=1 
and stops being above 15o on week 36.1 from the second equation when n=1.

Edwin

Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

1 year = 52 weeks approximately

Desmos Graph
https://www.desmos.com/calculator/4eidoto8nc

The Desmos graph shows the sine function f(x) = 20*sin( (pi/13)x ) + 2, along with the horizontal line y = 15.

On the domain 0 < x < 52, the intersection points are approximately:
A = (2.928,15)
B = (10.072,15)
C = (28.928,15)
D = (36.072,15)

The gap from A to B is one chunk of time when the temperature is above 15 degrees Celsius.
10.072-2.928 = 7.144, which is about 7 weeks.

The gap from C to D is another chunk of time when the temperature is above 15 degrees Celsius.
36.072-28.928 = 7.144, also about 7 weeks.

The total would then be roughly 7+7 = 14 weeks out of the 52 week timespan.

Edit: The student has mentioned the textbook's answer is 14.29
This is because 2*7.144 = 14.288 which rounds to 14.29