A musician is playing a tuba. If she plays a note and holds it for a significant length of time, the air pressure near the opening of the tuba can be modeled by the following function. pt=−101,325.8 - 0.1 cos163.1t
In this equation, pt represents the air pressure in pascals, and t is the time in seconds.
Find the following. If necessary, round to the nearest hundredth
Period of p :
Frequency of p:
Max air pressure:
We get the period from
cos163.1t
since period = 2π/k
period = 2π/163.1 = appr .0385 seconds
frequency: 163.1 cycles/second
The max of the cosine is +0.1
so the max of pt is -101325.8 + .1 = -101325.7
(btw, sounds like the musician is playing a third octave E)
To find the period of p, we need to determine the length of one complete cycle of the cosine function in the given equation.
The general form of a cosine function is: y = A cos(Bx)
In this case, the function is: pt = -101,325.8 - 0.1 cos(163.1t)
Comparing the equation to the general form, we can see that B = 163.1.
The period (P) of a cosine function is calculated using the formula: P = 2π/B
So, in this case, P = 2π/163.1.
To find the frequency (f) of p, we can use the formula: f = 1/P.
To find the maximum air pressure, we need to find the maximum value of the cosine function.
The maximum value of the cosine function is 1, so we can substitute that into the equation to find the maximum air pressure:
pt = -101,325.8 - 0.1(1)
pt = -101,325.8 - 0.1
pt = -101,325.9
Therefore, the period of p is approximately (2π/163.1) seconds, the frequency of p is approximately 1/(2π/163.1) Hz, and the maximum air pressure is approximately -101,325.9 pascals.
To find the period of a function, we need to find the value of t for which the cosine function completes one full cycle. The period, denoted as T, is the time it takes for the function to repeat its pattern.
In this case, we can see that the coefficient of t inside the cosine function is 163.1. The coefficient's value represents how the cosine function behaves over a specific range of t values. To find the period, we use the formula:
T = 2π / |coefficient of t|
In our equation, the coefficient of t is 163.1, so we can substitute this value into the formula:
T = 2π / 163.1
Calculating this expression, we find:
T ≈ 0.03854
Therefore, the period of the function is approximately 0.04 seconds (rounded to the nearest hundredth).
Next, let's find the frequency of the function. Frequency is the reciprocal of the period and represents the number of cycles completed per unit of time. Thus, we can use the formula:
Frequency = 1 / Period
Substituting the value of the period we found earlier:
Frequency ≈ 1 / 0.03854
Calculating this expression, we find:
Frequency ≈ 25.92
Therefore, the frequency of the function is approximately 25.92 cycles per second (rounded to the nearest hundredth).
Lastly, let's find the maximum air pressure. In the given function, the maximum air pressure occurs when the cosine function reaches its maximum value of 1. Therefore, we can simply evaluate the function at any value of t that corresponds to the maximum point.
Since the cosine function repeats every period, we can use t = 0 as a reference to find the maximum air pressure:
pt = -101,325.8 - 0.1 * cos(163.1 * 0)
Evaluating this expression, we find:
pt ≈ -101,325.7
Therefore, the maximum air pressure is approximately -101,325.7 pascals (rounded to the nearest hundredth).