Outside temperature over a day can be modelled as a sinusoidal function. Suppose you know the high temperature for the day is 87 degrees and the low temperature of 63 degrees occurs at 5 AM. Assuming t is the number of hours since midnight, find an equation for the temperature, D, in terms of t.

• So I am pretty sure the equation would start y=25sin(x) because 25 is the difference between the highest and lowest temperature
• In a sine equation the point at 3pi/2 is the lowest, so I think we have to move the graph over so that 5 is the lowest point??? Because the lowest temp happens at 5 hours past midnight. That doesn't really make sense but I'm not sure how to go about this problem.

2 A = 87-63 = 24

so A = 12 degrees
average = (87+63)/2 = 75
so
D = 75 + 12 sin (2 pi t/24 - p)
where p is the phase to give the right temp when t = 5
when t = 5
sin (2 pi 5/24 - p) = -1

or 5 pi/12 - p = -pi/2
p = 6 pi/12 + 5 pi/12 = 11 pi/12

so
D = 75 + 12 sin ( pi t/12 - 11 pi/12)
or
D = 75 + 12 sin [ pi/12(t-11) ]

Well, well, well... looks like the temperature is getting all sinusoidal on us! Don't worry, I'm here to clown around and help you out.

First, let's break down the problem. We know that the high temperature is 87 degrees and the low temperature of 63 degrees occurs at 5 AM. So, we have a temperature range of 87 - 63 = 24 degrees.

Now, let's focus on finding the equation for the temperature, D, in terms of t. We can start by using the basic form of a sine function: y = A sin(Bx - C) + D.

In this case, we already have the amplitude A (which is the temperature range) as 24. Let's plug that into the equation: y = 24 sin(Bx - C) + D.

Now, the next thing we need to figure out is the period of the sine function. The period is the length between two peaks or troughs. Since the temperature cycle repeats every 24 hours, the period is 24 hours. This means that B in our equation is equal to 2π/24, which simplifies to π/12.

Let's update our equation: y = 24 sin((π/12)x - C) + D.

Now, we just need to find the values of C and D to complete the equation. We know that at 5 AM (which is 5 hours past midnight), the temperature is at its lowest, which is 63 degrees. So, we substitute these values into our equation:

63 = 24 sin((π/12)(5) - C) + D.

Simplifying further, we have:

63 = 24 sin(5π/12 - C) + D.

Now, unfortunately, I can't directly solve for C and D as I can't crunch numbers, but with some math magic and clownery, you can solve it. Remember, C determines the horizontal shift and D determines the vertical shift of the graph.

So, put on your mathematical clown hat and solve for C and D, and then you can put it all together to get the final equation for the temperature, D, in terms of t.

I hope this helps in your temperature modeling adventure! It's quite a circus, isn't it?

To model the temperature as a sinusoidal function, we can start by considering the midpoint between the highest and lowest temperatures as the average temperature. In this case, the average temperature would be (87 + 63)/2 = 75 degrees.

Now, we need to determine the amplitude and period of the sinusoidal function. The amplitude is half the difference between the highest and lowest temperatures, which in this case is (87 - 63)/2 = 12 degrees.

The period of the sinusoidal function is the time it takes for it to complete one cycle. Since the low temperature occurs at 5 AM and the high temperature occurs sometime during the day, the period is 24 hours.

Finally, we need to determine the phase shift of the function, which represents any horizontal translations. In this case, the low temperature occurs at t = 5 which is 5 hours after midnight. Since the period is 24 hours, this means the phase shift is 5/24 of the period to the right.

Putting all this information together, we can write the equation for the temperature, D:
D = A*sin(2*pi*(t - c)/P) + M

Where:
A = amplitude = 12
P = period = 24
c = phase shift = 5/24
M = average temperature = 75

Therefore, the equation for the temperature, D, in terms of t is:
D = 12*sin(2*pi*(t - 5/24)/24) + 75

To model the temperature as a sinusoidal function, we can use the general equation for a sine function:

D = A * sin(B * (t - C)) + D

where:
- D is the temperature,
- A is the amplitude (half the difference between the high and low temperatures),
- B is the period (the number of hours it takes for the temperature to complete one cycle),
- C is the phase shift (the number of hours it is shifted to the right or left), and
- D is the vertical shift (the average temperature).

Given that the high temperature is 87 degrees and the low temperature of 63 degrees occurs at 5 AM, we can determine the values of A, B, C, and D.

1. Amplitude (A):
The difference between the high temperature and the low temperature is 87 - 63 = 24. Therefore, the amplitude is half of this difference: A = 24/2 = 12.

2. Period (B):
The period of the sine curve is the number of hours it takes to complete one full cycle. In a 24-hour day, there are 24 hours. Since the temperature cycle repeats every 24 hours, the period is 24.

3. Phase Shift (C):
The phase shift determines the horizontal shift of the graph. In this case, the low temperature occurs at 5 AM. This means there is a phase shift of 5 hours to the right (later in the day). Therefore, C = -5 (negative since it is a shift to the right).

4. Vertical Shift (D):
The average temperature is the midpoint between the high and low temperatures. Since 87 is the high temperature and 63 is the low temperature, the average temperature is located halfway between them: D = (87 + 63)/2 = 75.

Plugging these values into the equation, we get:

D = 12 * sin((2π/24) * (t - 5)) + 75

So the equation for the temperature, D, in terms of t is:

D = 12 * sin((π/12) * (t - 5)) + 75