Question

Measuring the rate of increase of blood alcohol concentration (BAC): Biomedical scientists have studied the chemical and physiological changes in the body that result from alcohol consumption. The reaction in the human body occurs in two stages: a fairly rapid process of absorption and a more gradual one of metabolism. To predict the effect of alcohol consumption, one needs to know the rate at which alcohol is absorbed and metabolized. Medical researchers measured the BAC of eight fasting adult male subjects after rapid consumption of 15 mL of ethanol (alcohol). The data they obtained were modeled by the concentration function $C(t) = 0.02te^{-0.05t}$ where $t$ is measured in minutes after consumption and $C$ is measured in mg/mL. Determine how quickly is the BAC increasing after 10 minutes? When is BAC maximum and how much it is? How rapidly was it decreasing half an hour later?

          Measuring the rate of increase of blood alcohol concentration (BAC):
Biomedical scientists have studied the chemical and physiological changes in the body that result from
alcohol consumption. The reaction in the human body occurs in two stages: a fairly rapid process of
absorption and a more gradual one of metabolism. To predict the effect of alcohol consumption, one
needs to know the rate at which alcohol is absorbed and metabolized.
Medical researchers measured the BAC of eight fasting adult male
subjects after rapid consumption of 15 mL of ethanol (alcohol). The
data they obtained were modeled by the concentration function
$C(t) = 0.02te^{-0.05t}$ where $t$ is measured in minutes after
consumption and $C$ is measured in mg/mL.
Determine how quickly is the BAC increasing after 10 minutes?
When is BAC maximum and how much it is? How rapidly was it
decreasing half an hour later?

Show more…

Measuring the rate of increase of blood alcohol concentration (BAC):
Biomedical scientists have studied the chemical and physiological changes in the body that result from
alcohol consumption. The reaction in the human body occurs in two stages: a fairly rapid process of
absorption and a more gradual one of metabolism. To predict the effect of alcohol consumption, one
needs to know the rate at which alcohol is absorbed and metabolized.
Medical researchers measured the BAC of eight fasting adult male
subjects after rapid consumption of 15 mL of ethanol (alcohol). The
data they obtained were modeled by the concentration function
C(t) = 0.02te^-0.05t where t is measured in minutes after
consumption and C is measured in mg/mL.
Determine how quickly is the BAC increasing after 10 minutes?
When is BAC maximum and how much it is? How rapidly was it
decreasing half an hour later?

Added by Tina P.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert H M

05/11/2023

Step 1

The given function is: c(t) = 0.02t * e^{-0.05t} Now, let's find the derivative of c(t) with respect to t: c'(t) = \frac{d}{dt}(0.02t * e^{-0.05t}) To find the derivative, we'll use the product rule: (uv)' = u'v + uv' u = 0.02t, u' = 0.02 v = e^{-0.05t}, v' = Show more…

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Measuring the rate of increase of blood alcohol concentration (BAC): Biomedical scientists have studied the chemical and physiological changes in the body that result from alcohol consumption. The reaction in the human body occurs in two stages: a fairly rapid process of absorption and a more gradual one of metabolism. To predict the effect of alcohol consumption, one needs to know the rate at which alcohol is absorbed and metabolized. Medical researchers measured the BAC of eight fasting adult male subjects after rapid consumption of 15 mL of ethanol (alcohol). The data they obtained were modeled by the concentration function C(t) = 0.02te^-0.05t, where t is measured in minutes after consumption and C is measured in mg/mL. Determine how quickly is the BAC increasing after 10 minutes? When is BAC maximum and how much it is? How rapidly was it decreasing half an hour later?

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Transcript

00:01 The bac is given by this formula.

00:06 To find how quickly the bac is increasing after 10 minutes, we need to find the derivative of c.

00:14 This tells us the rate of change of c.

00:17 I'm sorry, i left out a factor of t here.

00:25 This is 0 .02 t.

00:29 Now when we calculate the derivative, we'll use the product rule.

00:32 We have 0 .02 t times negative 0 .05 e to the negative 0 .05 t power, plus 0 .02 e to the negative 0 .05 t.

00:49 We can factor out 0 .02 e to the negative 0 .05 t, and we're left with negative 0 .05 t plus 1.

01:04 To find how quickly the bac is increasing after 10 minutes, we just plug in t equals 10.

01:17 We have 0 .02 e to the negative 0 .05 times 10 times negative 0 .05 times 10 plus 1.

01:33 This is 0 .02 e to the negative 0 .5 times negative 0 .5 plus 1, which is positive 0 .5, which simplifies to 0 .01 times e to the negative 0 .5.

01:54 This calculates to approximately 0 .00607, which is 6 .07 times 10 to the negative 3.

02:14 The units are milligrams per milliliter per minute.

02:23 This is the rate of increase of the bac after 10 minutes.

02:28 The bac is maximum when the derivative is 0.

02:39 Let's see when this is.

02:41 We have 0 equals 0 .02 times e to the negative 0 .05 t multiplied by negative 0 .05 t plus 1.

03:03 I can divide out these factors here.

03:08 I'm left with 0 equals negative 0 .05 t plus 1.

03:14 We have 0 .05 t equals 1, so t equals 20.

03:24 To determine if at this time bac is maximized rather than minimized, i'm going to do the second derivative test.

03:32 C double prime is equal to, we're using the product rule, so we get the derivative of negative 0 .05 t plus 1 is just negative 0 .05...

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