The maximum angle of the sun above the horizon for a small town in Ontario was recorded on the 21st of each month and is displayed in the table below. Create an equation to model the data.
Month
1
2
3
4
5
6
7
8
9
10
11
12
Angle
25
35
45
56
65
68
65
56
45
35
25
22
max-min = 40
average = (max+min)/2 = 45
so it looks kind of like
y = 45 - 20cos(2pi/12 x)
To create an equation that models the data, we need to find a relationship between the month and the maximum angle of the sun above the horizon.
From the given table, it can be observed that the angle increases from month 1 to month 6 and then decreases from month 7 to month 12. This indicates a periodic behavior.
It can also be noticed that the angle is symmetric about month 6. So, we can use a sine function that is centered around month 6 to model the data.
Let's define the angle (A) as a function of the month (M):
A(M) = A_max * sin(B * (M - M_0))
Where:
- A(M) is the maximum angle of the sun above the horizon for the given month.
- A_max is the maximum angle recorded in the table.
- B is the frequency or rate of change of the sine function.
- M_0 is the center month (in this case, month 6).
To find the values for A_max, B, and M_0, we can use the data provided for month 6 and month 12:
A(6) = A_max * sin(B * (6 - M_0)) = 68
A(12) = A_max * sin(B * (12 - M_0)) = 22
By evaluating these two equations simultaneously, we can find A_max, B, and M_0.
Let's solve these equations step-by-step:
Step 1: Divide the second equation by the first equation to eliminate A_max:
sin(B * (12 - M_0)) / sin(B * (6 - M_0)) = 22/68
Step 2: Use the trigonometric identity (sin(A) / sin(B) = (2 * cos((A + B)/2) * sin((A - B)/2)) to simplify the equation:
2 * cos(((12 - M_0) + (6 - M_0)) / 2) * sin(((12 - M_0) - (6 - M_0)) / 2) = 22/68
Step 3: Simplify the trigonometric expressions:
2 * cos(9 - M_0) * sin(3) = 22/68
Step 4: Divide both sides by 2 * sin(3):
cos(9 - M_0) = (11/34) * sin(3)
Step 5: Take the inverse cosine (arccos) of both sides:
9 - M_0 = arccos((11/34) * sin(3))
Step 6: Solve for M_0 by subtracting arccos((11/34) * sin(3)) from both sides:
M_0 = 9 - arccos((11/34) * sin(3))
Step 7: Substitute the value of M_0 into either of the original equations (A(6) = A_max * sin(B * (6 - M_0))) to find A_max:
A_max * sin(B * (6 - (9 - arccos((11/34) * sin(3))))) = 68
Step 8: Solve for A_max by dividing both sides by sin(B * (6 - (9 - arccos((11/34) * sin(3))))):
A_max = 68 / sin(B * (6 - (9 - arccos((11/34) * sin(3)))))
Overall, the equation to model the data is:
A(M) = (68 / sin(B * (6 - (9 - arccos((11/34) * sin(3)))))) * sin(B * (M - (9 - arccos((11/34) * sin(3)))))
Please note that the exact values for A_max, B, and M_0 can be calculated using numerical methods or a regression analysis of the data.
To create an equation to model the data, we need to determine the relationship between the month and the angle. One common way to model data like this is to use a trigonometric function.
In this case, the data appears to follow a pattern where the angle starts at a low value, increases to a maximum point, and then decreases back to a low value, repeating itself yearly. This suggests that a sine function would be a good fit.
The general equation for a sine function is given by:
y = A * sin(B(x - C)) + D
Where:
- A represents the amplitude, which is half the difference between the maximum and minimum values of the data.
- B represents the period, which is 2π divided by the length of one complete cycle of the data.
- C represents a phase shift, which determines the starting point of the data.
- D represents the vertical shift, which can move the entire graph up or down.
To determine the values of A, B, C, and D, we need to analyze the given data points.
Looking at the table, we can observe that the maximum angle occurs in months 3 and 7 (March and July), and the minimum angle appears in months 1 and 11 (January and November). This suggests a period of 12 months.
Using this information, we can determine the values:
- The amplitude (A) is half the difference between the maximum angle (68°) and the minimum angle (22°), which is (68 - 22) / 2 = 23°.
- The period (B) is 2π divided by the cycle length of 12 months, which is 2π / 12 = π/6 ≈ 0.524 radians.
- The phase shift (C) is 0 since the data starts in January.
- The vertical shift (D) is the average of the maximum and minimum angle, which is (68 + 22) / 2 = 45°.
Now we can write the equation to model the data:
y = 23 * sin((π/6)(x - 0)) + 45
Simplifying this equation further, we get:
y = 23 * sin(π/6 * x) + 45
Therefore, the equation to model the data is y = 23 * sin(π/6 * x) + 45.