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?
stuck please help :/
the period is 24, so
y = sin(π/12 x)
The temperature varies between 60 and 80, so the center-line is y=70, and the amplitude is 10
y = 10sin(π/12 x) + 70
I don't know what time the temperature is 70, but if it's at 6 am, then the horizontal shift is 6 hours, meaning
y = 10 sin(π/12 (x-6)) + 70
see
http://www.wolframalpha.com/input/?i=plot+y%3D10+sin(%CF%80%2F12+(x-6))+%2B+70,+y%3D70+for+0%3C%3Dx%3C%3D24
To find the equation for the sine function that models the situation, we need to determine the amplitude, period, and horizontal shift of the function.
The amplitude (A) of a sine function represents the maximum distance the curve varies from its average value. In this case, the temperature varies between 60°F and 80°F, so the amplitude is (80°F - 60°F)/2 = 10°F.
The period (P) of a sine function represents the length of one complete cycle. Since the given situation represents a 24-hour period, the period of the function is 24 hours.
The horizontal shift (C) represents any constant value added or subtracted from the input variable, in this case, time. Since the curve starts at 70°F in the morning, we need to shift the curve horizontally so that it starts at x = 0. This means the horizontal shift is 0.
Now we can write the equation for the sine function f(x):
f(x) = A * sin(2π/P * (x - C))
Substituting the values we found:
f(x) = 10 * sin(2π/24 * (x - 0))
Simplifying:
f(x) = 10 * sin(π/12 * x)
Therefore, the equation for the sine function that models the temperature over the 24-hour period is f(x) = 10 * sin(π/12 * x).
To model the temperature using a sinusoidal function, we can start by finding the amplitude, period, and horizontal shift.
1. Amplitude: The amplitude measures the distance from the midline to the maximum or minimum value of the function. In this case, the temperature rises 10°F above the midline (70°F) and falls 10°F below the midline, so the amplitude is 10.
2. Period: The period is the length of one complete cycle of the function. In this case, the temperature goes through one cycle from the morning to the next morning, which is 24 hours. So the period is 24.
3. Horizontal shift: Since the temperature starts at 70°F in the morning, we need to shift the graph horizontally so that the maximum value occurs at x = 0. The horizontal shift is given by x = -c, where c is the time when the maximum value occurs. In this case, the maximum value occurs when x = 12 (midday), so the horizontal shift is x = -12.
Now we can write the equation for the sinusoidal function f(x):
f(x) = A * sin(B(x - C)) + D
where:
A = amplitude = 10
B = 2π/period = 2π/24
C = horizontal shift = -12
D = vertical shift = midline value = 70
Putting it all together, the equation for the sine function f(x) is:
f(x) = 10 * sin[(2π/24)(x + 12)] + 70