Answer :
To determine which function best models the volume of air in Andy's lungs over time, we need to analyze the given data and understand how the lung volume varies with each breath.
1. Frequency of Breathing:
- Andy records 15 breaths per minute.
- This means the breathing rate, or cycle time, is [tex]\(60 / 15 = 4\)[/tex] seconds per breath.
2. Volume Inhaled/Exhaled per Breath:
- Andy inhales and exhales [tex]\(0.5\)[/tex] liters of air per breath.
- This means the volume change throughout the breathing cycle is from [tex]\(2\)[/tex] liters (residual) to [tex]\(2.5\)[/tex] liters (including the inhaled [tex]\(0.5\)[/tex] liters).
3. Residual Air and Amplitude:
- After exhaling, there are [tex]\(2\)[/tex] liters of residual air in Andy's lungs.
- The inhaled/exhaled volume of [tex]\(0.5\)[/tex] liters corresponds to an amplitude of [tex]\(0.25\)[/tex] liters (since [tex]\(\frac{0.5}{2} = 0.25\)[/tex]).
4. Waveform and Angular Frequency:
- We know the function should be a cosine function since the breathing starts at the beginning of an exhale (i.e., at maximum volume variation starting from the residual volume).
- The angular frequency ([tex]\(\omega\)[/tex]) is computed from the breathing rate:
[tex]\[ \text{Breath frequency} = \frac{1}{4 \text{ seconds}} = 0.25 \text{ Hz} \][/tex]
[tex]\[ \text{Angular frequency} (\omega) = 2 \pi \times 0.25 = \frac{\pi}{2} \text{ radians/second} \][/tex]
Now, let's compare each of the provided functions against these findings:
1. [tex]\( A(t) = 0.25 \cos \left( \frac{\pi}{2} t \right) + 2 \)[/tex]
- This function has amplitude [tex]\(0.25\)[/tex] and angular frequency [tex]\(\frac{\pi}{2}\)[/tex], which match our calculations.
- Also, the residual volume ([tex]\(2\)[/tex] liters) is correct.
2. [tex]\( A(t) = 0.25 \cos \left( \frac{\pi}{2} t \right) + 2.25 \)[/tex]
- While the amplitude and angular frequency are correct, the residual volume is incorrect (should be [tex]\(2\)[/tex] liters, not [tex]\(2.25\)[/tex]).
3. [tex]\( A(t) = 0.5 \cos \left( \frac{\pi}{4} t \right) + 2.25 \)[/tex]
- This function incorrectly assigns the amplitude ([tex]\(0.5\)[/tex] instead of [tex]\(0.25\)[/tex]) and angular frequency ([tex]\(\frac{\pi}{4}\)[/tex] instead of [tex]\(\frac{\pi}{2}\)[/tex]).
- Also, the residual volume is incorrect.
4. [tex]\( A(t) = 0.5 \cos \left( \frac{\pi}{4} t \right) + 2 \)[/tex]
- While the residual volume is correct, the amplitude and angular frequency do not match [tex]\(0.25\)[/tex] and [tex]\(\frac{\pi}{2}\)[/tex], respectively.
Conclusion:
From the provided functions, the one that correctly models the amount of air in Andy's lungs [tex]\(t\)[/tex] seconds after he started recording is:
[tex]\[ \boxed{ A(t) = 0.25 \cos \left( \frac{\pi}{2} t \right) + 2 } \][/tex]
1. Frequency of Breathing:
- Andy records 15 breaths per minute.
- This means the breathing rate, or cycle time, is [tex]\(60 / 15 = 4\)[/tex] seconds per breath.
2. Volume Inhaled/Exhaled per Breath:
- Andy inhales and exhales [tex]\(0.5\)[/tex] liters of air per breath.
- This means the volume change throughout the breathing cycle is from [tex]\(2\)[/tex] liters (residual) to [tex]\(2.5\)[/tex] liters (including the inhaled [tex]\(0.5\)[/tex] liters).
3. Residual Air and Amplitude:
- After exhaling, there are [tex]\(2\)[/tex] liters of residual air in Andy's lungs.
- The inhaled/exhaled volume of [tex]\(0.5\)[/tex] liters corresponds to an amplitude of [tex]\(0.25\)[/tex] liters (since [tex]\(\frac{0.5}{2} = 0.25\)[/tex]).
4. Waveform and Angular Frequency:
- We know the function should be a cosine function since the breathing starts at the beginning of an exhale (i.e., at maximum volume variation starting from the residual volume).
- The angular frequency ([tex]\(\omega\)[/tex]) is computed from the breathing rate:
[tex]\[ \text{Breath frequency} = \frac{1}{4 \text{ seconds}} = 0.25 \text{ Hz} \][/tex]
[tex]\[ \text{Angular frequency} (\omega) = 2 \pi \times 0.25 = \frac{\pi}{2} \text{ radians/second} \][/tex]
Now, let's compare each of the provided functions against these findings:
1. [tex]\( A(t) = 0.25 \cos \left( \frac{\pi}{2} t \right) + 2 \)[/tex]
- This function has amplitude [tex]\(0.25\)[/tex] and angular frequency [tex]\(\frac{\pi}{2}\)[/tex], which match our calculations.
- Also, the residual volume ([tex]\(2\)[/tex] liters) is correct.
2. [tex]\( A(t) = 0.25 \cos \left( \frac{\pi}{2} t \right) + 2.25 \)[/tex]
- While the amplitude and angular frequency are correct, the residual volume is incorrect (should be [tex]\(2\)[/tex] liters, not [tex]\(2.25\)[/tex]).
3. [tex]\( A(t) = 0.5 \cos \left( \frac{\pi}{4} t \right) + 2.25 \)[/tex]
- This function incorrectly assigns the amplitude ([tex]\(0.5\)[/tex] instead of [tex]\(0.25\)[/tex]) and angular frequency ([tex]\(\frac{\pi}{4}\)[/tex] instead of [tex]\(\frac{\pi}{2}\)[/tex]).
- Also, the residual volume is incorrect.
4. [tex]\( A(t) = 0.5 \cos \left( \frac{\pi}{4} t \right) + 2 \)[/tex]
- While the residual volume is correct, the amplitude and angular frequency do not match [tex]\(0.25\)[/tex] and [tex]\(\frac{\pi}{2}\)[/tex], respectively.
Conclusion:
From the provided functions, the one that correctly models the amount of air in Andy's lungs [tex]\(t\)[/tex] seconds after he started recording is:
[tex]\[ \boxed{ A(t) = 0.25 \cos \left( \frac{\pi}{2} t \right) + 2 } \][/tex]
