Two consecutive days of the week experience a maximum temperature of 90 degrees F at 1:00pm and a minimum temperature of 44 degrees F at 1:00am. Write a cosine equation measured in house that would model this behavior. Let x=0 represent 1:00pm on the first day.
since the range is 44-90, the amplitude is (90-44)/2 = 23
The period is 24 hours, so
t = 23cos(2pi/24 x)+67
t(0) = 23cos(0)+67 = 23+67 = 90
t(12) = 23cos(2pi/24 * 12)+67 = 23cos(pi)+67 = -23+67 = 44
To model this behavior using a cosine equation, we will consider the temperature as a function of time in hours. Let's break down the problem and find the values we need to construct the equation.
Let's assume the temperature follows a cosine pattern, where the maximum temperature occurs at 1:00 pm and the minimum temperature occurs at 1:00 am. We'll also assume that the temperature repeats every 24 hours since we are looking at two consecutive days.
From the information given, we know that the maximum temperature is 90 degrees F, which occurs at 1:00 pm or x = 0. We also know that the minimum temperature is 44 degrees F, which occurs at 1:00 am or x = 12.
To construct the cosine equation, we can start with the general form:
f(x) = A * cos(B * (x - C)) + D
A: Amplitude (half the difference between maximum and minimum)
B: Period (1 cycle / 24 hours)
C: Phase shift (how much the equation is shifted horizontally)
D: Vertical shift (average of the maximum and minimum)
Using this information, we can fill in the values:
A = (90 - 44) / 2 = 23
B = 2π / 24
C = 0 (Since x = 0 represents 1:00 pm on the first day)
D = (90 + 44) / 2 = 67
Now we can write the cosine equation:
f(x) = 23 * cos((2π / 24) * (x - 0)) + 67
Simplifying the equation further:
f(x) = 23 * cos((π / 12) * x) + 67
So, the cosine equation that models this temperature behavior in hours is:
f(x) = 23 * cos((π / 12) * x) + 67.