Trig Function?

Question

The desert temp, H, oscillates daily between 40 degrees F @ 5am and 80 degrees F @ 5pm. write a possible formula for H in terms of t, measured in hours from 5 am. Explain your answer please or at least how to get the answer.

Answer 1

Basically you are talking about a periodic quantity which repeats itself with time. Since the temperature is bounded (in your problem between 40 to 80) and has a period of 24h, and moreover is a differentiable variable every periodic differentiable function with 24h period and bounded whose graph passes (5,40) and (17,80), can be an answer. Maybe the easiest of all to check is a sine function. Let's pick a general sine function :

H = x*sin(at +b) + h is a good candidate, we need to find x, a,b and h. The maximum of this function happens when sin(at +b) equals to 1 which will be x+h the minimum comes at sin(at+b)=-1 where the minimum will be -x+h. Since we want the maximum temperature be 80 and minimum 40 we will have :
x+h = 80
-x+h = 40
thus h = 60 and x = 20

Since the period of sin(at +b) is 2*pi/a and we want that it be equal to 24 we have :
2*pi/a = 24 so a=pi/12

And finally we need to find b, it's easy just plug the temperature at a given time into your function H(t)=20*sin(t*pi/12 +b) + 60

H(5) = 20*sin(5*pi/12 +b) +60 = 40 which yields :

sin(5*pi/12 +b) = -1 or 5*pi/12 +b = 3*pi/2 so :
b= 13*pi/12 and finally this is your function :

>>>>> H(t)=20*sin(t*pi/12 + 13*pi/12) + 60 <<<<

G.L!

Answer 2

The function will be of the form

H = 60 + 20*sin(kt+ø)

when the value of sin is -1, H = 40, and when the value of sin is +1, H = 80, so the temp extremes are 40 and 80.

The 40 deg temp occurs at t = 5, which must make the sin = -1; we know that sin(3π/2) = -1, so

k*5+ø = 3*π/2

12 hrs later (5PM or 17:00 hrs) the value of sin should be +1, which occurs at sin(5π/2) so

k*17+ø = 5*π/2

You have two equation in two unknowns, k and ø:

subtract first from the second to get k*12 = π; then

k=π/12

Then use this in one of the original equations:

5*π/12 + ø = 3*π/2

which gives ø = 13*π/12

so the equation for the temperature is

H = 60 + 20*sin(π*t/12 + 13*π/12),

but t has to be on the 24-hr clock.

EDIT: by using trig identities, the same function can be expressed in many ways. Since sin(x-π/2) = cos(x) another answer is

H = 60 + 20*cos(π*t/12 + 7*π/12) = 60 + 20*cos[π/12*(t + 7)]

Answer 3

H = 60 + 20sin(15t - 90)