Outside temperature over a day can be modelled as a sinusoidal function. Suppose you know the temperature is 80 degrees at midnight and the high and low temperature during the day are 90 and 70 degrees, respectively. Assuming t is the number of hours since midnight, find an equation for the temperature, D, in terms of t.

Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the high temperature of 52 degrees occurs at 6 PM and the average temperature for the day is 45 degrees. Find the temperature, to the nearest degree, at 8 AM.

let's pick a sine function

the range is 9-70 = 20
so a = 10
we know that the period is 24 hrs
so 2π/k=24
24k = 2π
k = π/12

so far we have:
D = 10 sin(π/12)(t) + 80
giving us a range from 70 to 90

But obviously the temp after midnight would decrease, whereas our function has it increasing to 90 when t = 6 (6:00 am)
We could do a phase shift, or more simply, just flip the function to
D = -10sin(π/12)t + 80

check some values
t = 0 , D = -10sin0 + 80 = 80 , ok
t=6 , (6:00 am) D = -10 sin (π/2) + 80 = 70 , ok
t = 12 (noon), D = -10 sin π + 80 = 80 , ok
t = 18 , (6:00 pm) , D = -10sin 3π/2 + 80 = 90

all looks good

Well, when it comes to modeling temperature as a sinusoidal function, things can get pretty "heated"! But don't worry, I'm here to help you cool things down with a dose of mathematical humor.

Let's start with some given information. At midnight, the temperature is a cool 80 degrees. Later in the day, it reaches a high of 90 degrees and drops down to a low of 70 degrees. We can use these values to create an equation for the temperature, D, in terms of time, t.

Now, we know that the sine function oscillates between -1 and 1, so we'll need to adjust the amplitude and scale. In this case, let's say the amplitude is (90 - 70) / 2 = 10, and the scale is 12 hours in a day.

Using this information, our equation for the temperature, D, in terms of time, t, would be:

D = 10 * sin((2π/12) * t) + 80

So, next time someone asks you about temperature modeling, you can impress them with this "chill" equation! Stay cool!

To find an equation for the temperature, D, in terms of time, t, we can use a sinusoidal function. The general form of a sinusoidal function is:

D = A * sin(Bt + C) + D0

Where:
A is the amplitude (half the difference between the maximum and minimum values)
B determines the period of the function (the number of hours it takes for one complete cycle)
C is the phase shift (how the function is shifted horizontally)
D0 is the vertical shift (the midline of the function)

In this problem, we have the following information:
- The maximum temperature is 90 degrees, which will be the amplitude A.
- The minimum temperature is 70 degrees.
- The temperature at midnight is 80 degrees, which will be the vertical shift D0.
- The period of the function is 24 hours, as it is a daily cycle.

Step 1: Finding the amplitude (A)
The amplitude is half the difference between the maximum and minimum values:
A = (90 - 70) / 2 = 20 / 2 = 10

Step 2: Finding the vertical shift (D0)
The vertical shift is the temperature at midnight:
D0 = 80

Step 3: Finding the period (B)
The period is the length of one complete cycle, which is 24 hours:
B = 2π / 24 = π / 12

Therefore, the equation for the temperature, D, in terms of time, t, will be:

D = 10 * sin((π / 12) * t) + 80

This equation represents the sinusoidal variation of temperature throughout the day, with an amplitude of 10, a vertical shift of 80, and a period of 24 hours.

Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the temperature is 55 degrees at midnight and the high and low temperature during the day are 74 and 36 degrees, respectively. Assuming t is the number of hours since midnight, find an equation for the temperature, D, in terms of t.