A musician is playing a tuba. If he 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.
p(t)= 101,325.8- 0.1 cos 163.1t
In this equation, p(t) represents the air pressure in pascals, and t is the time in seconds.
Find the following. If necessary, round to the nearest hundredth.
Time for one full cycle of p: __ seconds
Frequency of p: __ cycles per second
Minimum air pressure: __ pascals
mathhelper did this for you 3 hours ago
check your previous post
To find the time for one full cycle of p, we need to find the period of the function. The period is the time it takes for the function to complete one full cycle.
In this case, the function is in the form p(t) = A + B * cos(C * t), where A, B, and C are constants. The period of this function can be found using the formula T = 2π / C.
In our equation, C = 163.1, so the period T is given by:
T = 2π / 163.1 ≈ 0.0385 seconds.
Therefore, the time for one full cycle of p is approximately 0.0385 seconds.
To find the frequency of p, we can use the formula f = 1 / T, where f represents the number of cycles per second.
In this case, the frequency f is given by:
f = 1 / 0.0385 ≈ 25.97 cycles per second.
Therefore, the frequency of p is approximately 25.97 cycles per second.
To find the minimum air pressure, we need to determine the minimum value of the function p(t). In this case, the minimum value occurs when cos(163.1t) is equal to -1, since the range of cosines is from -1 to 1.
So, p(t) = 101,325.8 - 0.1 * (-1) = 101,325.9 pascals.
Therefore, the minimum air pressure is approximately 101,325.9 pascals.
To find the time for one full cycle of p, we need to determine the period of the cosine function. The period of a cosine function is determined by the coefficient of t inside the cosine function.
In this case, the coefficient is 163.1. The period is given by the formula:
Period = 2π / |coefficient|
So, the period of our cosine function is:
Period = 2π / 163.1
Calculating this value, we get:
Period ≈ 0.03857 seconds
Therefore, the time for one full cycle of p is approximately 0.03857 seconds.
To find the frequency of p, we can use the formula:
Frequency = 1 / Period
So, the frequency of p is:
Frequency ≈ 1 / 0.03857
Calculating this value, we get:
Frequency ≈ 25.92 cycles per second
Therefore, the frequency of p is approximately 25.92 cycles per second.
To find the minimum air pressure, we need to find the minimum value of p(t). The minimum value of a cosine function is found when the cosine function is at its maximum magnitude, which is -1.
So, we need to find the maximum magnitude of the cosine function in our equation. The coefficient of the cosine function, in this case, is 0.1. The maximum magnitude is given by the formula:
Maximum Magnitude = |coefficient|
In our case, the maximum magnitude is:
Maximum Magnitude = |0.1|
Therefore, the minimum air pressure is approximately 0.1 pascals.