.Over a 24-hour period, the temperature in a town can be modeled by one period of a sinusoidal function. The temperature measures 70°F in the morning, rises to a high of 80°F, falls to a low of 60°F, and then rises to 70°F by the next morning.
What is the equation for the sine function f(x), where x represents time in hours since the beginning of the 24-hour period, that models the situation?
To find the equation for the sine function that models the temperature in the town over a 24-hour period, we need to determine the important features of the function.
1. Amplitude: The amplitude represents the difference between the highest and lowest points of the function. In this case, the temperature rises to a high of 80°F and falls to a low of 60°F, so the amplitude is half the difference between these temperatures, which is (80 - 60) / 2 = 10°F.
2. Midline: The midline represents the average value of the function. The temperature measures 70°F in the morning and rises to 70°F by the next morning. So, the midline is 70°F.
3. Period: The period represents the length of one complete cycle of the function. In this case, the period is 24 hours since it represents a full day.
Now we can write the equation for the sine function:
f(x) = A * sin(B(x - C)) + D
where:
- A is the amplitude
- B is the frequency (2π divided by the period)
- C is the horizontal shift (a phase shift)
- D is the vertical shift (the midline)
Using the given information, we can plug in the values:
A = 10
B = 2π / 24 = π / 12
C = 0 (since there is no horizontal shift)
D = 70 (since the midline is at 70°F)
Therefore, the equation for the sine function f(x) that models the temperature is:
f(x) = 10 * sin((π/12)x) + 70