The volume of air contained in the lungs of a certain athlete is modeled by the equation v=447sin(76πt)+721, where t is time in minutes, and v is volume in cubic centimeters.
What is the maximum possible volume of air in the athlete's lungs?
What is the minimum possible volume of air in the athlete's lungs?
How many breaths does the athlete take per minute?
ive tried this 3 times now and cant seem to get it
Answer??
To find the maximum and minimum possible volume of air in the athlete's lungs, we need to identify the highest and lowest values of the equation.
1. Maximum Possible Volume:
The equation is in the form v = Asin(Bt) + C, where A is the amplitude, B is the frequency, and C is the vertical shift. In this case, A = 447, B = 76π, and C = 721.
The maximum value of the sine function is 1. Therefore, the maximum possible volume can be calculated by adding the amplitude to the vertical shift:
Maximum Volume = Amplitude + Vertical Shift
= 447 + 721
= 1168 cubic centimeters
2. Minimum Possible Volume:
The minimum volume occurs when the sine function reaches its lowest value, which is -1. Therefore, we need to subtract the amplitude from the vertical shift to find the minimum possible volume:
Minimum Volume = Vertical Shift - Amplitude
= 721 - 447
= 274 cubic centimeters
3. Breaths per Minute:
To find the number of breaths the athlete takes per minute, we need to calculate the time it takes for one complete cycle of the sine function. The period (T) of the sine function is given by T = 2π/B.
B = 76π
T = 2π / 76π
= 2/76
= 1/38
Therefore, the athlete completes one breath cycle every 1/38 minutes or approximately 1.58 minutes.
To find the number of breaths per minute, we can take the reciprocal of the period:
Breaths per Minute = 1 / (1/38)
= 38 breaths per minute
Hence, the maximum possible volume of air in the athlete's lungs is 1168 cubic centimeters, the minimum possible volume is 274 cubic centimeters, and the athlete takes approximately 38 breaths per minute.
To find the maximum and minimum volumes of air in the athlete's lungs, we can analyze the equation v = 447sin(76πt) + 721.
1. Maximum volume:
The maximum value of sin function is 1. Therefore, we need to find the maximum value of the expression 447sin(76πt), and then add 721 to it.
To find the maximum value of sin(76πt), we can set 76πt = π/2, which corresponds to the maximum value of 1 for sin(x).
So, solve for t:
76πt = π/2
t = (π/2) / (76π)
t = 1 / (152)
Substitute this value of t back into the equation:
v = 447sin(76π(1/152)) + 721
v = 447sin(π/2) + 721
v = 447(1) + 721
v = 447 + 721
v = 1168
Therefore, the maximum possible volume of air in the athlete's lungs is 1168 cubic centimeters.
2. Minimum volume:
The minimum value of sin function is -1. Similar to finding the maximum value, we need to find the minimum value of the expression 447sin(76πt) and then add 721 to it.
To find the minimum value of sin(76πt), we can set 76πt = 3π/2, which corresponds to the minimum value of -1 for sin(x).
So, solve for t:
76πt = 3π/2
t = (3π/2) / (76π)
t = 1 / (152/3)
t = 3 / 152
Substitute this value of t back into the equation:
v = 447sin(76π(3/152)) + 721
v = 447sin(3π/2) + 721
v = 447(-1) + 721
v = -447 + 721
v = 274
Therefore, the minimum possible volume of air in the athlete's lungs is 274 cubic centimeters.
3. Breaths per minute:
To determine the number of breaths the athlete takes per minute, we need to analyze the frequency of the sinusoidal function.
The general equation for a sine function is y = A*sin(Bx + C) + D, where A is the amplitude and B affects the frequency.
From the given equation, we have sin(76πt). The coefficient of t, which is 76π, determines the frequency.
In this case, we have a frequency of 76π breaths per minute.
So, the athlete takes 76π breaths per minute.