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.
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a) To determine the function for the temperature with respect to the number of days since Jan. 1st, we can use a sinusoidal function. Let's start by finding the amplitude and period.
The amplitude is the absolute value of half the difference between the highest temperature (27 degrees Celsius) and the lowest temperature (-11 degrees Celsius). So the amplitude is (27 - (-11))/2 = 38/2 = 19 degrees Celsius.
The period is the number of days it takes to complete one full cycle of temperature change. In this case, it's the number of days between two consecutive hottest or coldest days, which corresponds to one year. Since there are 365 days in a year, the period is 365 days.
Now, we can use the general form of a sinusoidal function:
f(x) = A * sin(2π/T * (x - C)) + D
where A is the amplitude, T is the period, C is the horizontal shift (which represents the day the function starts), and D is the vertical shift (which represents the average temperature).
In this case, A = 19, T = 365, C = 0 (since we start counting from Jan. 1st), and D is the average temperature, which is the midpoint between the highest and lowest temperatures. So D = (27 + (-11))/2 = 16/2 = 8 degrees Celsius.
Now we have our function:
f(x) = 19 * sin(2π/365 * x) + 8
b) To determine the temperature on July 6th, we need to find the number of days between Jan. 1st and July 6th. Since Jan. 1st is day 1, July 6th is day 187.
Plugging that into our function:
f(187) = 19 * sin(2π/365 * 187) + 8
We can evaluate this expression to find the temperature on July 6th.
Similarly, to determine the temperature on Nov. 16th, we need to find the number of days between Jan. 1st and Nov. 16th. Jan. 1st is day 1, and Nov. 16th is day 320.
Plugging that into our function:
f(320) = 19 * sin(2π/365 * 320) + 8
Evaluate this expression to find the temperature on Nov. 16th.