The formula modeling the temperature is f(t) = 15 cos(2πt / 24) + 55.
Analyzing the Graph:
The graph you sent shows the temperature of the surface water in a lake over a 24-hour period. Here are some key observations we can make from the graph:
- Periodic behavior: The temperature fluctuates up and down in a repeating pattern, indicating a periodic behavior. This suggests that the temperature can be modeled by a sinusoidal function.
- Amplitude: The temperature varies between approximately 40°F and 70°F, with a difference of 30°F. This represents the amplitude of the sinusoidal function.
- Period: The temperature completes one full cycle within 24 hours. This is the period of the sinusoidal function.
- Midline: The average temperature across the cycle is (40°F + 70°F) / 2 = 55°F. This represents the midline of the sinusoidal function.
Modeling the Temperature with a Sinusoidal Function:
The general form of a sinusoidal function that can model the temperature is:
f(t) = A cos(B(t - C)) + D
where:
- A is the amplitude, representing the half-distance between the maximum and minimum values.
- B is the angular frequency, determining the period of the oscillation.
- C is the horizontal shift, indicating the phase displacement.
- D is the midline, representing the average value.
Matching the Function to the Graph:
In this case, we can determine the specific values of the parameters in the function to match the graph:
- Amplitude (A): 15°F (half of the temperature range)
- Period (B): 2π/24 radians per hour (since the period is 24 hours)
- Phase shift (C): 0 hours (since the maximum temperature occurs at t = 0)
- Midline (D): 55°F (the average temperature)
Therefore, the specific formula that models the surface temperature of the lake is:
f(t) = 15 cos(2πt / 24) + 55
This formula captures the periodic behavior, amplitude, period, and midline observed in the graph.