1. In 2001, Windsor, Ontario received its maximum amount of sunlight, 15.28 hrs, on June 21, and its least amount of sunlight, 9.08 hrs, on December 21. On what day(s) can Windsor expect 13.5 hours of sunlight?
I determined the equation for this is D(h) = 3.1cos[360/365(h – 172)] + 12.2. I know the answer is 238 and 106, but was only able to get 238 as a solution:
3.1cos[360/365(h – 172)] + 12.2 = 13.5
3.1cos[360/365(h – 172)] = 1.3
cos[360/365(h – 172)] = 1.3/3.1
360/365(h – 172) = arccos(1.3/3.1)
360/365(h – 172) = 65.20613754
h – 172 = 365/360(65.20613754)
h – 172 = 66.11177834
h = 238.1117783 = 238
I was able to get 106 as a solution by doing the following:
238 – 172 = 66
172 – 66 = 106
What is the proper way of getting 106 as the second solution?
2. A mass suspended on a spring will exhibit sinusoidal motion when it moves. If the mass on a spring is 85 cm off the ground at its highest position and 41 cm off the ground at its lowest position and takes 3.0 s to go from the top to the bottom and back again, determine an equation to model the data.
I was able to determine the following:
A = 22
B = 120
D = 63
So H(t) = 22sin[120(h – C) + 63
I was unable to determine the C value, but by plugging in this formula into graphing software and playing around the numbers, I got 2.25 as the C value, but not sure how to get this answer from the given data…
To determine the proper way of getting 106 as the second solution in question 1, let's revisit the equation:
cos[360/365(h – 172)] = 1.3/3.1
To find the second solution, we need to take the inverse cosine (arccos) of the right side:
arccos(1.3/3.1) = 65.20613754
Now, to find the second solution, we subtract this value from 360:
360 - 65.20613754 = 294.7938625
Then, we solve for h by adding 172 to both sides:
h = 294.7938625 + 172 = 466.7938625
However, since we are interested in the day number, we need to convert this to an integer value. Rounding down the result, we get:
h = 466
Now, let's move on to question 2.
To determine the equation to model the data for the mass suspended on a spring, we can start by considering the given information:
Highest position: 85 cm off the ground
Lowest position: 41 cm off the ground
Time taken from top to bottom and back: 3.0 s
From this, we can determine the amplitude (A), which is half the difference between the highest and lowest positions:
A = (85 - 41)/2 = 22 cm
Next, we can determine the period (T), which is the time taken for one complete cycle of the sinusoidal motion:
T = 2 * 3.0 s = 6.0 s
To find the angular frequency (ω), we can use the formula ω = 2π/T:
ω = 2π/6.0 = π/3 rad/s
Now, we can construct the equation for the sinusoidal motion, which will be of the form H(t) = A*sin(ωt + φ), where φ represents the phase shift.
Since the mass is at its highest position when t = 0, we can determine the phase shift:
H(0) = 22*sin(π/3 * 0 + φ) = 85 cm
Sinus(φ) = 85/22
φ ≈ 2.25 rad
Therefore, the equation to model the data is:
H(t) = 22*sin((π/3)t + 2.25)
This equation represents the height (H) of the mass above the ground as a function of time (t).