The temperature, throughout the year, in a particular town can be graphed using trigonometric functions. On July 24th, the hottest day of the year, the temperature is 27 degrees Celsius. On Jan. 24th, the coldest day of the year, the temperature is -11 degrees Celsius.
Question 4 options:
a)
Determine the function for the temperature with respect to the number of days since Jan. 1st. [Hint: Jan. 1st is day 1]
b)
Determine what the temperature will be on July 6th and Nov. 16th.
a) To determine the function for the temperature with respect to the number of days since Jan. 1st, we can use a cosine function to model the temperature variation throughout the year. The general form of a cosine function is given as:
T(t) = A * cos(B(t + C)) + D
where:
- T(t) represents the temperature at time t
- A represents the amplitude of the function (half the difference between the maximum and minimum temperatures)
- B represents the period of the function (the length of time it takes to complete one full cycle)
- C represents the phase shift of the function
- D represents the vertical shift of the function
Given that the hottest day of the year occurs on July 24th (day 205) with a temperature of 27 degrees Celsius, and the coldest day of the year occurs on Jan. 24th (day 24) with a temperature of -11 degrees Celsius, we can use this information to determine the values of A, B, C, and D.
First, we calculate the amplitude (A):
A = (27 - (-11)) / 2
A = 38 / 2
A = 19
Next, we calculate the period (B):
We know that the period of a cosine function is equal to 365 days (or one year). Therefore, B = (2π) / 365.
Now, for the phase shift (C), we need to find the value of t that corresponds to Jan. 1st. Since Jan. 1st is day 1, we can directly use this value.
Finally, the vertical shift (D) is equal to the average of the maximum and minimum temperatures:
D = (27 + (-11)) / 2
D = 16 / 2
D = 8
Putting it all together, the function for the temperature (T) with respect to the number of days (t) since Jan. 1st is:
T(t) = 19 * cos((2π / 365)(t + 1)) + 8
b) To determine the temperature on July 6th, we need to find the value of t that corresponds to that date. Since July 6th is 187 days after Jan. 1st, we can substitute t = 187 into the function above and calculate the temperature:
T(187) = 19 * cos((2π / 365)(187 + 1)) + 8
Similarly, to determine the temperature on Nov. 16th, we substitute t = 320 into the function:
T(320) = 19 * cos((2π / 365)(320 + 1)) + 8
By evaluating these equations, you can find the temperatures on July 6th and Nov. 16th.
a) To determine the function for the temperature with respect to the number of days since Jan. 1st, we can use a combination of sine and cosine functions since they have periodic behavior.
Let's start by finding the period of the function. The period is the number of days it takes for the temperature pattern to repeat. In this case, the period is 365 days because it takes one year for the temperature to cycle through all the seasons.
Next, we need to find the amplitude of the function. The amplitude is the maximum deviation from the average temperature. In this case, the hottest day of the year is 27 degrees Celsius, and the coldest day is -11 degrees Celsius. The average temperature can be found by taking the midpoint between these two extremes:
Average temperature = (27 + (-11))/2 = 8 degrees Celsius
So, the amplitude is the difference between the hottest day and the average temperature:
Amplitude = 27 - 8 = 19 degrees Celsius
Now, we can construct the equation for the temperature as a function of the number of days since Jan. 1st. Let's call the number of days x.
The general formula for a sinusoidal function is:
T(x) = A * sin(2π*(x - x0)/P) + C
where:
- A is the amplitude (in this case, 19)
- x0 is the phase shift (the starting point of the sinusoidal function)
- P is the period (365 days)
- C is the vertical shift (the average temperature, 8 degrees Celsius)
Since Jan. 1st is considered day 1, we know the temperature on Jan. 24th (day 24) is -11 degrees Celsius. This gives us a starting point to determine the phase shift (x0).
x0 = 24 - 1 = 23
Now we can put all the values together and determine the function for the temperature:
T(x) = 19 * sin(2π*(x - 23)/365) + 8
b) To find the temperature on July 6th and Nov. 16th, we need to substitute the corresponding values for x in the equation we just derived.
For July 6th (day 187), we substitute x = 187 into the equation:
T(187) = 19 * sin(2π*(187 - 23)/365) + 8
For Nov. 16th (day 320), we substitute x = 320 into the equation:
T(320) = 19 * sin(2π*(320 - 23)/365) + 8
Calculating these equations will give you the temperature on those specific days.
the range is from -11 to 27 or 38°
so in our trig function we have
T = 19sin(.....) + 8 as a start
the period is 365 days, so 2π/k = 365
and k = 2π/365
so we have T = 19 sin (2π/365)d + 8
where d is the number of the day in our year
(there are charts available that will give you the day number for a given date, e.g March 3 = 63)
In its current form our trig function will give up a max of 27 and a min of -11 , but for the wrong dates, so we need a phase shift
how about R = 19 sin (2π/365)(d +k)) + 8
check: for Jan 24 , d = 24, T = -11
-11 = 19sin(2π/365)(24+k) + 8
-19 = 19sin(2π/365)(24+k)
sin(2π/365)(24+k) = -1
I know sin (3π/2) = -1
(2π/365)(24+k) = 3π/2
24 + k = (3π/2)(365/2π) = 1095/4
k = 999/4
so T = 19 sin(2π/365)(d+999/4) + 8
checking it for jan24 or d = 24
my calculator gives me: T = -11 , YEAHHHH
checkin go July 24 , d = 205 ?? (check that)
T = 26.9936° , well, how about that
so for july 6, I get d = 187
T = 19 sin(2π/365)(187+999/4) + 8
= 25.94°
You do the one for Nov 16, happy button-pushing
(make sure you start with the innermost brackets, and follow the order of operation. Repeat my calculations first to make sure you get the same results. )