.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?
70 is the midpoint (average of high and low), so we will have
f(x) = 70+asin(kx)
the amplitude is 10, so
f(x) = 70+10sin(kx)
The period is 24 hours, so 2pi/k=24
f(x) = 70+10sin(pi/12 x)
To find the equation for the sine function that models the given temperature situation, we need to determine the amplitude, period, phase shift, and vertical shift.
Amplitude: The amplitude is half the difference between the maximum and minimum values of the temperature. In this case, the amplitude would be (80°F - 60°F) / 2 = 10°F.
Period: The period is the length of one complete cycle. In this case, the temperature reaches its maximum value at the high point and then returns to the same temperature 24 hours later. Therefore, the period would be 24 hours.
Phase Shift: The phase shift determines the horizontal translation of the graph. The temperature is at its maximum in the middle of the 24-hour cycle. Since the temperature measures 70°F in the morning and reaches its maximum at the high point, the phase shift would be 12 hours.
Vertical Shift: The vertical shift determines how the graph is shifted up or down. In this case, the temperature starts at 70°F and drops to 60°F before rising to its maximum. Therefore, the vertical shift would be (70°F + 60°F) / 2 = 65°F.
Putting it all together, the equation for the sine function f(x) would be:
f(x) = 10*sin((2π/24)*(x - 12)) + 65
This equation represents the temperature, where x represents the time in hours since the beginning of the 24-hour period.
To find the equation for the sinusoidal function f(x) that models the temperature over a 24-hour period, we need to identify the key features of the function.
From the given information, we know that the temperature starts at 70°F in the morning, rises to a high of 80°F, falls to a low of 60°F, and then rises back to 70°F by the next morning. This indicates that the midline (average temperature) of the function is 70°F, and the amplitude (difference between the midline and the maximum or minimum temperature) is (80°F - 70°F) / 2 = 5°F.
Since we want the function to complete one period over a 24-hour period, the period is 24 hours. The equation for a sinusoidal function can be written as:
f(x) = A * sin(B * (x - C)) + D
where:
A is the amplitude
B is the horizontal stretch/shrink factor
C is the phase shift
D is the vertical shift
From our given information, we can determine the values of A, B, C, and D.
A = 5 (the amplitude)
B = 2π / T, where T is the period. In our case, T = 24 hours. So B = 2π / 24 = π / 12.
C indicates the phase shift. Since the temperature starts at 70°F in the morning, there is no phase shift, so C = 0.
D represents the vertical shift, which is equal to the midline value. In this case, D = 70°F.
Now we can substitute these values into the equation to get the final equation for the sinusoidal function f(x):
f(x) = 5 * sin((π / 12) * (x - 0)) + 70
Simplifying it further, we get:
f(x) = 5sin(πx / 12) + 70
Therefore, 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 is:
f(x) = 5sin(πx / 12) + 70