The number of hours of daylight in a city in the northern hemisphere shows periodic behavior over time.
- The average number of daylight hours is 12
- The maximum number of daylight hours is 14.4
- The period is 365 days
- The day with the least sunlight is december 20
which equation models the number of hours of daylight on the day that comes t days after the shortest day of the previous year?
A) H(t)= -14.4 sin(0.017t)
B) H(t)= 14.4 sin(0.017t)
C) H(t)= -2.4 sin(0.017t) + 12
D) H(t)= 2.4 sin(0.017t) + 12
the amplitude is the max - average = 2.4
so, start out with
H(t) = 12+2.4sin(kt)
the period is 265, so k = 2pi/365 = 0.017
H(t) = 12+2.4sin(0.017t)
since the minimum is at t=0, we want
H(t) = 12-2.4cos(0.017t)
Not a choice, unless there is a typo and you want (C)
Yes, I'm very sorry, the last two are meant to be cos, and I was debating whether it was C or D, thank you for your help!:)
What was the answer to this question?
C) H(t)= -2.4 sin(0.017t) + 12
Let me shed some light on this question with a little humor.
The equation that models the number of hours of daylight on the day that comes t days after the shortest day of the previous year is option D) H(t) = 2.4 sin(0.017t) + 12.
Now, why did I choose this option? Well, let's break it down:
- The sine function helps capture the periodic behavior beautifully, like waves crashing on the beach. So, we definitely need that sine in there.
- Since we're looking for the number of hours after the shortest day, we need to add 12, because you know, we don't want to subtract daylight. It's precious!
- And the 2.4 coefficient before the sine gives us just the right range of hours, keeping things realistic. We don't want to go overboard and have 24-hour days (unless we're in the land of the midnight sun, of course).
So the equation H(t) = 2.4 sin(0.017t) + 12 is just shining with the right balance of humor and mathematical accuracy.
To answer this question, we need to consider the given information about the number of hours of daylight in a city in the northern hemisphere.
We know that the period is 365 days, which means the pattern repeats every year. The day with the least sunlight is December 20, and we need to find the equation that models the number of hours of daylight on the day that comes t days after the shortest day of the previous year.
Since the average number of daylight hours is 12 and the maximum number is 14.4, we can assume that the function representing the hours of daylight will be a sinusoidal curve oscillating between these two values.
The general equation for a sinusoidal function is of the form:
H(t) = A * sin(B * (t - C)) + D
Where:
- A is the amplitude of the function (half the difference between the maximum and minimum values)
- B affects the period of the function (related to the length of the period)
- C is the horizontal shift (how far the function is shifted along the x-axis)
- D is the vertical shift (the average value of the function)
From the given information, we can determine the values needed in the equation:
- The amplitude A is half the difference between the maximum and minimum values: (14.4 - 12) / 2 = 1.2
- The period is 365 days, so B = 2π / 365 (since 2π is a full revolution)
- The horizontal shift C is the number of days between December 20 and the start of the year (January 1), so C = 12.
- The vertical shift D is the average number of daylight hours, which is 12.
Therefore, the equation that models the number of hours of daylight on the day t days after the shortest day of the previous year is:
H(t) = 1.2 * sin((2π / 365) * (t - 12)) + 12
Simplifying further, we have:
H(t) = 1.2 * sin(0.017t - 0.066) + 12
Comparing this equation to the given options, we can see that the correct choice is:
D) H(t) = 2.4 * sin(0.017t) + 12