Problem 70 - Calculus for Biology and Medicine

Question

You are measuring the ability of an antibiotic to kill harmful bacteria. You measure the rate at which the antibiotic kills bacteria (i.e., number of bacteria killed in one hour); this is called the mortality rate. You measure the following data for the number of bacteria killed in a 12 hour time period starting at \(t=0\), and ending at \(t=12\). $$ \begin{array}{c|c} \hline \text { Time, } t & \text { Mortality rate, per hour } \boldsymbol{m}(\boldsymbol{t}) \\ \hline 0 & 20 \\ 1 & 300 \\ 2 & 350 \\ 3 & 400 \\ 4 & 500 \\ 5 & 450 \\ 6 & 410 \\ 7 & 350 \\ 8 & 320 \\ 9 & 300 \\ 10 & 200 \\ 11 & 100 \\ 12 & 110 \\ \hline \end{array} $$ (a) Use six even subintervals to approximate the total number of deaths between \(t=0\) and \(t=6\) and evaluate this sum using the data in the table. (b) Use six even subintervals to approximate the total number of deaths between \(t=0\) and \(t=12\) and evaluate this sum using the data in the table. (c) Use four even subintervals to approximate the total number of deaths between \(t=4\) and \(t=12\) and evaluate this sum using the data in the table.

Step-by-Step Solution

Answer

(a) 2020 deaths. (b) 3800 deaths. (c) 2400 deaths.

Step 1: Define the Interval for Part (a)

For part (a), we need to approximate the total number of deaths from \(t=0\) to \(t=6\) using six even subintervals. Each subinterval will therefore be 1 hour long (as the interval is 6 hours).

Step 2: Calculate the Mortality within Each Subinterval for Part (a)

Using the given mortality rates, calculate the number of deaths for each subinterval from time \(t=0\) to \(t=6\). The subintervals are: [0,1], [1,2], [2,3], [3,4], [4,5], and [5,6]. Hence, using Midpoint Rule for simplicity, we take the mortality rate at the midpoint of each subinterval and multiply it by the interval length (1 hour).- [0,1]: Mortality \(= 20 \,/\, hour\)- [1,2]: Mortality \(= 300 \,/\, hour\)- [2,3]: Mortality \(= 350 \,/\, hour\)- [3,4]: Mortality \(= 400 \,/\, hour\)- [4,5]: Mortality \(= 500 \,/\, hour\)- [5,6]: Mortality \(= 450 \,/\, hour\)Then total deaths approx = \(20 + 300 + 350 + 400 + 500 + 450\).

Step 3: Sum the Mortality Rates for Part (a)

Adding the mortality rates for each subinterval we get:\[ 20 + 300 + 350 + 400 + 500 + 450 = 2020 \]Thus, the total number of deaths between \(t=0\) and \(t=6\) is approximately 2020.

Step 4: Define the Interval for Part (b)

For part (b), we approximate the total number of deaths from \(t=0\) to \(t=12\) using six even subintervals, each of 2 hours (since 12/6=2).

Step 5: Calculate the Mortality within Each Subinterval for Part (b)

For subintervals of length 2 hours each, pick mortality rates at midpoints:- [0,2]: Midpoint at \(t=1\), Mortality \(= 300\)- [2,4]: Midpoint at \(t=3\), Mortality \(= 400\)- [4,6]: Midpoint at \(t=5\), Mortality \(= 450\)- [6,8]: Midpoint at \(t=7\), Mortality \(= 350\)- [8,10]: Midpoint at \(t=9\), Mortality \(= 300\)- [10,12]: Midpoint at \(t=11\), Mortality \(= 100\)Then total deaths approx = \(2 imes (300 + 400 + 450 + 350 + 300 + 100)\).

Step 6: Sum the Mortality Rates for Part (b)

Calculate the total deaths by evaluating the sum:\[ 2 imes (300 + 400 + 450 + 350 + 300 + 100) = 3800 \]Thus, the total number of deaths between \(t=0\) and \(t=12\) is approximately 3800.

Step 7: Define the Interval for Part (c)

For part (c), approximate the total number of deaths from \(t=4\) to \(t=12\) using four even subintervals, each of 2 hours (since 8/4=2).

Step 8: Calculate the Mortality within Each Subinterval for Part (c)

For subintervals of length 2 hours each, consider mortality at midpoints:- [4,6]: Midpoint at \(t=5\), Mortality \(= 450\)- [6,8]: Midpoint at \(t=7\), Mortality \(= 350\)- [8,10]: Midpoint at \(t=9\), Mortality \(= 300\)- [10,12]: Midpoint at \(t=11\), Mortality \(= 100\)Then total deaths approx = \(2 imes (450 + 350 + 300 + 100)\).

Step 9: Sum the Mortality Rates for Part (c)

Calculate the total number of deaths by evaluating the sum:\[ 2 imes (450 + 350 + 300 + 100) = 2400 \]Thus, the total number of deaths between \(t=4\) and \(t=12\) is approximately 2400.

Key Concepts

Midpoint Rule

The Midpoint Rule is a numerical integration technique used in calculus to approximate the area under a curve. Specifically, it breaks down the area into multiple subintervals and then uses the midpoint of each subinterval to estimate the value of the function. This simplifies the integration process by providing an average value that reflects the function's behavior over each subinterval.

To apply the Midpoint Rule, divide the interval into equal parts. For each part, identify the midpoint and use the function's value at that midpoint. Multiply this value by the width of each subinterval to calculate the approximate contribution to the integral. For example, if the interval is from 0 to 6, divided into six subintervals, the midpoint of the first subinterval [0,1] is 0.5. Measure the function value at time 0.5, and this approximates the behavior of the function between 0 and 1.

Using the Midpoint Rule helps in approximating the integral without performing complex mathematical analysis, making it a convenient method especially when working with tabulated data like mortality rates.

Mortality Rate Calculation

In biological contexts, mortality rate refers to the number of individuals dying per unit time. It's a crucial measure in studies related to population health and the effectiveness of medical treatments, like antibiotics.

Mortality rate calculation involves understanding not just how many individuals die over a set period but breaking this period into smaller intervals. Each interval's death count can be analyzed to understand changing patterns over time. For a study over a 12-hour window, mortality rates at specific hours provide insight into how the antibiotic affects bacteria, where peaks could indicate times of maximum bacterial vulnerability.

This measurement is not only vital for evaluating treatment effectiveness but is also foundational in creating models for disease progression and control.

Subintervals in Calculus

Subintervals are smaller sections of a larger interval that are used to simplify complex calculus problems. When you partition an interval into subintervals, you aim to make calculations more manageable.

Choosing the size of these subintervals is important. Equally spaced subintervals, commonly used in applications like the Midpoint Rule, provide a uniform approach to evaluation. In the example from the exercise, different interval endpoints like [0,1], [1,2], etc., break the time duration into manageable parts to facilitate calculation of bacteria deaths per hour.

This approach is not unique to mortality rate studies; it's common across many fields of calculus. Understanding subintervals helps students tackle not just integration problems but any phenomena modeled by calculus.

Numerical Integration

Numerical integration is a method for approximating the value of an integral. In situations where an analytical solution is not feasible, numerical methods, like the Midpoint Rule or Trapezoidal Rule, come into play.

For scientists and engineers, numerical integration allows complex systems modeled by data points to be analyzed effectively. In the calculation of bacteria mortality rates over time, numerical integration helps in summarizing the total bacterial deaths based on the changing mortality rates.

While exact values are approached through traditional integration, numerical methods offer practical solutions when working with discrete data points or non-integrable functions. They are essential in converting real-world observations into workable mathematical results efficiently.