Find the amplitude, frequency, and distance when t=3
a. y=-15cos((pi*t)/3)+5
b. y=120sin100pi*t
I think the amp for a. is 15, the freq is 3/2 cycles per unit of time, and the distance is 20, but I'm not totally sure if I'm approaching this problem correctly.
(a)
The period is (2pi) / (pi/3) = 6
so, the frequency is 1/6
your amplitude and distance are correct.
Remember, the period for cos(kt) is 2pi/k
So, (b) should be no trouble now.
Why isn't it (2pi) / (3pi) or (2pi) / (3), since the problem asks when t=3?
the period and amplitude do not change, whatever the value of t. The question asks for three things:
amplitude
frequency
distance when t=3
The distance when t=3 is
y(3) = -15cos((pi*3)/3)+5
= -15cos(pi)+5
= 15+5
= 20
To find the amplitude, frequency, and distance at a specific point in time, you can analyze the given equations. Let's start with equation (a):
a. y = -15cos((πt)/ 3) + 5
Amplitude: The amplitude represents the maximum value the function reaches from its midline (average value). In this case, the amplitude is the absolute value of the coefficient of the cosine function. So, the amplitude of equation (a) is |(-15)| = 15.
Frequency: The frequency determines how many times the function completes a full cycle within a given unit of time. To find the frequency, you need to look at the coefficient of "t" in the argument of the cosine function (πt/3). In this case, the coefficient is (π/3).
The frequency can be calculated as ω = 2πf, where ω is the angular frequency and f is the frequency. From the equation (πt/3), we can deduce that the angular frequency ω = π/3. Thus, the frequency is f = ω / (2π) = (π/3) / (2π) = 1/6.
Distance: To find the distance when t = 3, substitute t = 3 into the equation and calculate y.
y = -15cos((π*3)/3) + 5
= -15cos(π) + 5
= -15(-1) + 5
= 20
Therefore, the amplitude is 15, the frequency is 1/6 cycles per unit of time, and the distance when t = 3 is 20 for equation (a).
Moving on to equation (b):
b. y = 120sin(100πt)
Amplitude: Similar to equation (a), the amplitude is determined by the coefficient in front of the sine function. In this case, the amplitude is |120| = 120.
Frequency: To find the frequency, you need to look at the coefficient of "t" in the argument of the sine function (100πt). Here, the coefficient is 100π.
The angular frequency ω = 100π, and using the formula f = ω / (2π), we can calculate the frequency:
f = (100π) / (2π) = 100 / 2 = 50
Distance: To find the distance when t = 3, substitute t = 3 into the equation and calculate y:
y = 120sin(100π*3)
= 120sin(300π)
Without a specific value for π, the exact distance cannot be determined. However, we can conclude that the distance will lie between -120 and 120, depending on the value of π.
To summarize, the amplitude is 120, the frequency is 50 cycles per unit of time, and the distance when t = 3 for equation (b) is between -120 and 120.