The percentage of the moon's surface that is visible to a person standing on the Earth varies with the time since the moon was full. The moon passes through a full cycle in 28 days, from full moon to full moon. The maximum percentage of the moon's surface that is visible is 50%. Determine an equation, in the form P = A cos (Bt) + C for the percentage of the surface that is visible, P, as a function of the number of days, t, since the moon was full. Show the work that leads to the values of A, B, and C.
Well, it seems like you want me to do some math, but I prefer juggling numbers rather than solving equations. But hey, I'll give it a shot!
Since the moon goes through a full cycle in 28 days, we can assume that the percentage of the moon's surface visible is a function of time, t, in days. We are given that the maximum percentage visible is 50%.
From this information, we can determine the amplitude, A, of the cosine function. The amplitude is half the difference between the maximum and minimum values of the function. In this case, the maximum value is 50% and the minimum value must be 0%. Thus, the amplitude, A, is (50-0)/2 = 25.
The period, T, of the function is the time it takes for the function to complete one full cycle. In this case, the moon completes its cycle in 28 days, so the period, T, is 28.
Now we can determine the value of B, which stands for the frequency of the cosine function. The frequency is the reciprocal of the period, so B = 1/T = 1/28.
Lastly, we need to find C, the vertical shift or the phase offset of the function. Since the minimum value is 0%, we know that C = 0.
Now we have all the values. Plugging them into the equation P = A cos (Bt) + C, we get:
P = 25 cos((2π/28)t) + 0.
So there you have it, the equation that represents the percentage of the moon's surface visible as a function of the number of days since the moon was full. Just remember, I make jokes, not solve math equations!
To determine the equation in the form P = A cos(Bt) + C for the percentage of the moon's surface that is visible, we need to analyze the given information.
We know that the moon passes through a full cycle in 28 days, from full moon to full moon. During this cycle, the maximum percentage of the moon's surface that is visible is 50%.
Therefore, we can use the cosine function to model the change in visibility over time.
From the given information, we can deduce the following:
- The range of the cosine function is from -1 to +1, which corresponds to a percentage range of 0% to 100%. However, since the maximum visible percentage is only 50%, we can scale the cosine function down to a maximum amplitude of 0.5 (half of the full range).
- The period of the cosine function is 28 days, which corresponds to a full cycle of the moon.
Now, let's determine the values of A, B, and C:
A (amplitude): The maximum amplitude is given as 0.5, since the maximum visible percentage is 50%.
B (angular frequency): The period (T) of the cosine function is 28 days. The angular frequency (w) is calculated using the formula w = 2π / T.
w = 2π / 28 = π / 14
Since the general form of the cosine function is cos(Bt), we have B = π / 14.
C (vertical shift): The vertical shift represents the percentage of the moon's surface that is not visible. Since the maximum visible percentage is 50%, the vertical shift is 0.5 (half of the full range).
Putting it all together, the equation representing the percentage of the moon's surface that is visible as a function of the number of days since the moon was full is:
P = 0.5 cos(πt/14) + 0.5
To determine the equation for the percentage of the surface of the moon that is visible as a function of the number of days since the moon was full, we can use a cosine function. The general form of the equation is P = A cos(Bt) + C, where P is the percentage of the moon's surface visible, t is the number of days since the moon was full, and A, B, and C are constants that we need to determine.
Given that the maximum percentage visible is 50%, we know that the range of the cosine function is from -50% to +50%. Therefore, the amplitude A of the cosine function is 50%/2 = 25%.
Next, we need to determine the period of the function, which is the time it takes for the moon to complete a full cycle from full moon to full moon. We are given that the moon passes through a full cycle in 28 days. The period of a cosine function is equal to 2π divided by the value of B in the equation. So, we calculate B as follows:
Period = 2π / B
28 days = 2π / B
B = 2π / 28
B = π / 14
Now we have the amplitude A = 25% and the period B = π / 14. The last constant we need to determine is C, which represents a possible shift or offset of the cosine function.
To find C, we need to consider when the moon was last at full and whether it was at a maximum or minimum visibility. Given that the maximum percentage visible is 50%, we can assume the last full moon was at maximum visibility. Therefore, C is equal to 50%.
Now we have all the constants: A = 25%, B = π / 14, and C = 50%. Substituting these values into the equation P = A cos(Bt) + C, we get the final equation:
P = 25% cos((π/14)t) + 50%
This equation represents the percentage of the moon's surface that is visible as a function of the number of days since the moon was full.