Problem 32 - Applied Calculus

Question

Table \(1.19\) shows the concentration, \(c\), of creatinine in the bloodstream of a dog. \(^{40}\) (a) Including units, find the average rate at which the concentration is changing between the (i) \(6^{\text {th }}\) and \(8^{\text {th }}\) minutes. (ii) \(8^{\text {th }}\) and \(10^{\text {th }}\) minutes. (b) Explain the sign and relative magnitudes of your results in terms of creatinine.$$ \begin{array}{l} \text { Table } 1.19\\\ \begin{array}{c|ccccc} \hline t \text { (minutes) } & 2 & 4 & 6 & 8 & 10 \\ \hline c(\mathrm{mg} / \mathrm{ml}) & 0.439 & 0.383 & 0.336 & 0.298 & 0.266 \\\ \hline \end{array} \end{array} $$

Step-by-Step Solution

Answer

(i) -0.019 mg/ml per minute (ii) -0.016 mg/ml per minute

Step 1: Identify Data Points for Interval 6th to 8th Minute

For the interval between the 6th and 8th minutes, note the concentration at the 6th minute as \( c_6 = 0.336 \, \text{mg/ml} \) and at the 8th minute as \( c_8 = 0.298 \, \text{mg/ml} \).

Step 2: Calculate Average Rate of Change for 6th to 8th Minute

The formula for average rate of change is \( \frac{c_8 - c_6}{8 - 6} \). Substituting the known values: \( \frac{0.298 - 0.336}{8 - 6} = \frac{-0.038}{2} = -0.019 \, \text{mg/ml per minute} \).

Step 3: Identify Data Points for Interval 8th to 10th Minute

For the interval between the 8th and 10th minutes, note the concentration at the 8th minute as \( c_8 = 0.298 \, \text{mg/ml} \) and at the 10th minute as \( c_{10} = 0.266 \, \text{mg/ml} \).

Step 4: Calculate Average Rate of Change for 8th to 10th Minute

Using the formula for the average rate of change: \( \frac{c_{10} - c_8}{10 - 8} \). Substituting the known values: \( \frac{0.266 - 0.298}{10 - 8} = \frac{-0.032}{2} = -0.016 \, \text{mg/ml per minute} \).

Step 5: Interpret the Sign and Magnitude of Results

The negative signs indicate that the concentration of creatinine is decreasing over both intervals. The average rate of change is larger in magnitude between the 6th and 8th minutes (-0.019) compared to the 8th and 10th minutes (-0.016), suggesting a faster decrease in the earlier interval.

Key Concepts

Creatinine Concentration

Creatinine is a waste product that is produced by muscles during the breakdown of creatine. It is typically removed from the bloodstream by the kidneys, and its concentration in blood can indicate how well the kidneys are functioning. In this exercise, the creatinine concentration is measured in milligrams per milliliter (mg/ml) at different time intervals to observe changes over time. By tracking the creatinine concentration, we can analyze how quickly or slowly the kidneys are processing this waste. A steady decrease may indicate normal function, whereas inconsistencies might suggest potential issues with kidney performance.

Interval Analysis

Analyzing intervals involves examining the changes in creatinine concentration over specific time frames given in the problem: from the 6th to the 8th minute, and from the 8th to the 10th minute. In our context, this involves calculating the average rate of change of the creatinine concentration over these time spans.

• For the 6th to the 8th minute: Concentration drops from 0.336 mg/ml to 0.298 mg/ml.
• For the 8th to the 10th minute: Concentration further decreases from 0.298 mg/ml to 0.266 mg/ml.

These changes are found using the formula for the average rate of change, which is \( rac{c_2 - c_1}{t_2 - t_1} \). A negative result signals a decrease in concentration in each interval.

Mathematical Interpretation

Mathematically, the average rate of change provides insight into how quantities vary over a particular interval. In this case, the formula
\[ \text{Average Rate of Change} = \frac{c_{end} - c_{start}}{t_{end} - t_{start}} \] was employed to find the rate of decrease in creatinine concentration.

• From the 6th to the 8th minute: The average rate of change is -0.019 mg/ml per minute.
• From the 8th to the 10th minute: The average rate of change is -0.016 mg/ml per minute.

These negative rates signify a reduction in creatinine concentration. The more negative the rate, the quicker the concentration decreases. Between the 6th and 8th minutes, the concentration decreases faster compared to the 8th to 10th minutes. This information helps in understanding how the concentration evolves over time.