Answer:
2a: amount = 24.9(442/249)^(4t-1)
2b: 14.03 billion
2c: 1.38 trillion
2d: 1.158 hours
2e: 0.302 hours
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3a: 2200 mL
3b: 2450 mL
3c: 500 mL
3d: 5 s
3e: 1/5 Hz or 12 cycles per minute
Step-by-step explanation:
2.
(a) I find it easiest to write an exponential equation as follows:
define ...
a0 = initial given amount (not necessarily at time 0)
t0 = time corresponding to a0
a1 = second given amount (at a later time)
t1 = time corresponding to a1
write the equation as ...
amount = a0 · (a1/a0)^((t-t0)/(t1-t0))
This is usually a good place to start. It can be converted to the form ...
amount = a0·e^(kt)
if necessary or desirable.
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In terms of the above, the problem statement tells us ...
a0 = 24.9 billion
t0 = 1/4 hour
a1 = 44.2 billion
t1 = 1/2 hour
so our exponential equation is ...
amount = 24.9 · (44.2/24.9)^((t-1/4)/(1/2-1/4))
amount = 24.9(442/249)^(4t-1)
Please be aware there are many other ways to write this. Writing the equation this way will make it match the given numbers exactly. Most other forms will be an approximation. For example, you could write ...
amount ≈ 14.03(9.9287^t)
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(b) Filling in t=0, we find the amount to be ...
amount = 24.9(442/249)^-1 = 24.9(249/442) ≈ 14.0274 . . . . billion
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(c) Filling in t=2, we find the amount to be ...
amount = 24.9(442/249)^(4·2-1) = 24.9(442/249)^7 ≈ 1382.8 . . . billion
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(d) We want to find t for amount = 200. Filling in the number, we get ...
200 = 24.9(442/249)^(4t-1)
200/24.9 = (442/249)^(4t-1) . . . . . . divide by 24.9
log(200/24.9) = (4t -1)log(442/249) . . . . take logarithms
log(200/24.9)/log(442/249) = 4t -1 . . . . . divide by log(442/249)
Finishing the solution for t, we have ...
t = (1/4)(log(200/24.9)/log(442/249) +1) ≈ 1.1577 . . . hours
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(e) When td is added to t, the value of the exponential formula is doubled:
24.9(442/249)^(4(t+td)-1) = 2(24.9(442/249)^(4t-1)
Dividing by half the expression on the left side, we have ...
(442/249)^(4td) = 2
Taking logarithms, we get ...
4t·log(442/249) = log(2)
and dividing by the coefficient of t gives ...
t = log(2)/(4·log(442/249)) ≈ 0.3019686 . . . hours
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3. This question involves figuring out the amplitude and average value of a sine function. The amplitude is the multiplier of sin( ), so is 250 (mL). The average value is the value added to the sin( ) function, so is 2450 (mL).
At the maximum of the sine function, the amplitude is added to the average value. At the minimum, the amplitude is subtracted from the average value.
(a) The minimum lung capacity is 2450 -250 = 2200 mL.
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(b) The average lung capacity is 2450 mL.
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(c) The tidal volume is the difference between the maximum and minimum lung capacity. It is twice the amplitude, so is 2·250 mL = 500 mL.
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(d) The period of the function is the value of t that makes the argument of the sine function be 2π. That value is t=5. The period is 5 seconds. This is the time required for one full cycle of respiration.
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(e) The frequency in Hz is the reciprocal of the period in seconds. It is ...
1/(5 s) = 1/5 Hz
In the context of this problem, it means 1/5 of a cycle of respiration is completed each second. (In terms of cycles per minute, it is 12 cycles per minute. That is, the owner of these lungs completes about 12 full cycles of respiration per minute.)
