okay sorry last two questions! I promise!
The sun always illuminates half of the moon’s surface, except during a lunar eclipse. The illuminated portion of the moon visible from Earth varies as it revolves around Earth resulting in the phases of the moon. The period from a full moon to a new moon and back to a full moon is called a synodic month and is 29 days, 12 hours, and 44.05 minutes long. Write a sine function that models the fraction of the moon’s surface which is seen to be illuminated during a synodic month as a function of the number of days, d, after a full moon. [Note: full moon equals illuminated.]
I am really confused on this one I know I have to make a sine function.
so would it be like y= 29sin(pi/6*t)+44.05
sorry that was probably a really bad guess.
2)A bicycle tire has a diameter of 20 inches and is revolving at a rate of 10 rpm. At t = 0, a certain point is at height 0. What is the height of the point above the ground after 20 seconds?
y=20sin(pi/5*20) Is that right?
period=29 days, 12hrs, 44.05 min.
ok, change that to days: 29+12/24+44.05/(60*24), I get 29.53 days.
y=.5cos(2PI/29.53 * t)+.5 check it. Graph it.
2. it goes up and down over a 20 inch altitude. at t=0, it is at zero.
h=-10cos(2PI*10t)+10 is one solution, there are others, depending on where you call h=0. Height at ground level is not always called zero, you could call zero at the top.
1) To model the fraction of the moon's surface that is illuminated during a synodic month as a function of the number of days, d, after a full moon, we can use a sine function. The period of the function corresponds to the length of a synodic month, which is 29 days, 12 hours, and 44.05 minutes. To convert this to days, we can use:
Period = 29 days + (12 hours + 44.05 minutes) / 24 hours
Now, let's create the equation:
y = sin(2πd / Period)
Substituting the value of the period:
y = sin(2πd / (29 + 12/24 + 44.05/(24*60)))
2) The height of a point on a bicycle tire above the ground can be represented by a sine function. In this case, the diameter of the tire is 20 inches, so the radius is 10 inches. The tire is revolving at a rate of 10 revolutions per minute, which means it completes 10 cycles in 1 minute or 1 cycle in 6 seconds.
The formula for the height of the point on the tire above the ground can be expressed as:
y = R * sin(2πt / Period)
where R is the radius of the tire and Period is the time it takes for one complete revolution.
Substituting the given values:
y = 10 * sin(2πt / (1/10))
To answer your first question, let's break it down step by step.
The period of a synodic month is given as 29 days, 12 hours, and 44.05 minutes. In terms of hours, this is approximately 708.75 hours.
Next, we need to determine the fraction of the moon's surface illuminated at any given time, based on the number of days after a full moon. This fraction varies sinusoidally, so we can use a sine function to model it.
The formula for a sine function is:
y = A * sin(B * x)
In this case, we know that the period of the synodic month is 708.75 hours. So, for the function to complete one full cycle in this period, the coefficient B should be:
B = 2π / (708.75)
Now, let's determine the amplitude A. The function should range from 0 to 1, as it represents the fraction of the moon's surface illuminated. So, the maximum value of the sine function occurs at π/2, corresponding to a full moon (i.e., an illumination of 1). Thus, A = 1.
Putting it all together, the sine function that models the fraction of the moon's surface illuminated during a synodic month as a function of the number of days, d, after a full moon is:
y = sin((2π / 708.75) * d)
Now, let's move on to your second question.
The height of a point on the bicycle tire can be modeled by a sine function as it revolves. The circumference of the tire is given by π times the diameter, which in this case is 20 inches. So, the formula for the height y of the point above the ground at time t is:
y = A * sin(B * t)
The period of one revolution of the tire can be calculated by dividing 60 (to convert seconds to minutes) by the given angular velocity of 10 rpm. This gives us a period of 6 seconds.
Now, let's determine the amplitude A. The point starts at height 0, reaches a maximum height of the radius (half the diameter, which is 10 inches), and then returns to height 0. So, the amplitude A should be 10.
Finally, we can determine the coefficient B using the period of 6 seconds:
B = 2π / (6)
Putting it all together, the sine function that models the height of the point above the ground after 20 seconds is:
y = 10 * sin((2π / 6) * 20)
I hope this helps clarify how to approach these questions! Let me know if you have any further questions.