Question

Medicine The concentration \(M\) (in grams per liter) of a six-hour allergy medicine in a body is modeled by \(M=12-4 \ln \left(t^{2}-4 t+6\right), 0 \leq t \leq 6\), where \(t\) is the time in hours since the allergy medication was taken. Use Simpson's Rule with \(n=6\) to estimate the average level of concentration in the body over the six- hour period.

Short answer

To compute the average level of concentration of the medicine in the body over six hours, one should identify the parameters needed for the Simpson's rule given the problem details, apply Simpson's Rule to estimate the integral representing the total concentration over the given period of time, and then divide this integral by the length of the time period to get the average concentration.

Step-by-step solution

Identify the values

To apply the formula for Simpson's Rule given above, the values for \(\Delta x\), \(n\), and \(f(x)\) need to be identified. For this problem, given that \(0 \leq t \leq 6\) and \(n = 6\), \(\Delta x = (b - a) / n = (6 - 0) / 6 = 1\). The function \(f(x)\) is given by the concentration model \(M(t)\).

Apply Simpson's Rule

Applying Simpson's Rule to the model \(M(t)\), we compute the value of the integral \(\int_{0}^{6} M(t) dt\). Using Simpson's Rule, it is:\[S = \frac{1}{3} \left[M(0) + 4 \sum_{i=1}^{3} M(2i-1) + 2 \sum_{i=1}^{2} M(2i) + M(6)\right],\]where the values for \(M(t)\) are computed using the expression for \(M(t)\) given.

Estimate the average concentration

To find the average concentration over the six hour period, divide the integral computed in step 2 above by the length of the interval, i.e., 6 hours. This results in the formula \(\bar{M} = S / 6\).

Key concepts

Average Concentration

In the context of this exercise, average concentration refers to the mean level of concentration of a substance (in this case, the allergy medication) in the body over a specific period.
To estimate the average concentration using integration, first, you must calculate the total concentration over the specified time by integrating the concentration function. This gives you the cumulative effect of the variable concentration across time.
After obtaining the total concentration (which is done by using Simpson's Rule), the final step is to divide this value by the total time period, which is 6 hours in this case. This division yields the average concentration, offering an understanding of the medication's typical presence in the body over the duration.
Remember, average concentration is key in understanding how long and how effectively the medicine remains active over time.

Logarithmic Functions

Logarithmic functions, like the one in the concentration model, play a significant role in many natural phenomena, including medicine. In this exercise, the concentration model is given by the function \(M(t) = 12 - 4 \ln\left(t^2 - 4t + 6\right)\). Here, \(\ln\) represents the natural logarithm, which is the logarithm to base \(e\).
The presence of a logarithmic function could indicate that the rate of change in concentration slows down as time progresses, typical for many pharmacokinetic processes. The logarithm effectively compresses values, turning multiplicative changes in inputs into additive ones in the output, which helps to model processes where changes impact gradually.
Understanding this function helps in interpreting how the medication's concentration changes over time and how effectively it is delivered and absorbed by the body.

Numerical Integration

Numerical integration is a vital technique in approximating the area under curves represented by functions. In many cases, exact integration is not feasible, especially when dealing with complex or unknown functions. This is where numerical methods, such as Simpson's Rule, become crucial.
Simpson's Rule is a popular technique because it provides a good balance between accuracy and simplicity of computation. It divides the total interval into smaller segments, applies a quadratic polynomial approximation for each segment, and then sums up the contributions from each approximation. This rule is particularly applicable when the function is reasonably smooth, as it reduces errors significantly compared to simpler methods like the Trapezoidal Rule.
By using numerical integration, particularly Simpson's Rule, one can estimate the definite integral of the concentration function, which is necessary to determine the average concentration over the given period in the exercise.