Problem 1 - Calculus for Biology and Medicine

Question

Body mass index (or BMI) is often used as an indicator for whether a person is over- or underweight. A person's BMI is calculated from their mass (in \(\mathrm{kg}\) ) and their height (in \(\mathrm{m}\) ). To calculate a person's BMI, divide their mass by the square of their height. (a) If a person's mass is \(m\), and their height is \(h\), write down the formula that would be used to calculate their BMI. (b) Jesse is \(1.75 \mathrm{~m}\) tall, and he weighs \(82 \mathrm{~kg}\). What is his BMI? (c) In a particular population, heights range from \(1.50 \mathrm{~m}\) to \(1.90 \mathrm{~m}\), and masses range from \(45 \mathrm{~kg}\) to \(160 \mathrm{~kg} .\) Calculate the maximum possible range of BMI's for this population.

Step-by-Step Solution

Answer

Jesse's BMI is approximately 26.8. The BMI range for the population is from 12.5 to 71.1.

Step 1: Formula for BMI

To calculate the Body Mass Index (BMI) of a person, the formula used is: \[ \text{BMI} = \frac{m}{h^2} \] where \( m \) is the mass in kilograms (kg) and \( h \) is the height in meters (m).

Step 2: Calculate Jesse's BMI

Jesse's mass is \( 82 \) kg and his height is \( 1.75 \) m. Plug these values into the formula: \[ \text{BMI} = \frac{82}{(1.75)^2} \] Calculate \((1.75)^2\): \(1.75 \times 1.75 = 3.0625\). So, \[ \text{BMI} = \frac{82}{3.0625} \approx 26.8 \]

Step 3: Determine maximum BMI for the population

The maximum BMI occurs when the mass is at its maximum and the height is at its minimum. Using the values 160 kg (mass) and 1.50 m (height), the BMI is: \[ \text{Max BMI} = \frac{160}{(1.50)^2} \] Calculate \((1.50)^2\): \(1.50 \times 1.50 = 2.25\). So, \[ \text{Max BMI} = \frac{160}{2.25} \approx 71.1 \]

Step 4: Determine minimum BMI for the population

The minimum BMI occurs when the mass is at its minimum and the height is at its maximum. Using the values 45 kg (mass) and 1.90 m (height), the BMI is: \[ \text{Min BMI} = \frac{45}{(1.90)^2} \] Calculate \((1.90)^2\): \(1.90 \times 1.90 = 3.61\). So, \[ \text{Min BMI} = \frac{45}{3.61} \approx 12.5 \]

Key Concepts

Mathematical Formula for BMI

Body Mass Index, commonly known as BMI, is a simple yet effective way to assess whether a person's weight falls within a healthy range based on their height. To determine BMI, the following mathematical formula is used:
\[ \text{BMI} = \frac{m}{h^2} \]
In this formula, \( m \) represents the mass of an individual measured in kilograms, and \( h \) symbolizes height in meters. This formula essentially divides the mass by the square of the height. This means, effectively, that the distribution of body mass is being normalized over the body's height, providing a single numerical value representative of body fat for most people.

• Easy calculation: Requires just mass and height.
• Global applicability: Uses metric units that are standard worldwide.

Step-by-Step Calculation

Walking through the calculation step-by-step can demystify how the BMI formula works.Let's consider Jesse's example, who weighs 82 kg and is 1.75 meters tall. Here's how you'd calculate his BMI:1. **Compute the height squared**: First, find the square of Jesse’s height: - \((1.75)^2 = 3.0625\) This step involves multiplying Jesse's height by itself.2. **Substitute the values in the formula**: Insert Jesse’s weight and the squared height into the BMI formula: - \[ \text{BMI} = \frac{82}{3.0625} \]3. **Perform the division**: Divide Jesse's mass by the square of his height to find his BMI: - \[ \text{BMI} \approx 26.8 \]This precise, step-by-step method ensures clarity on how each component of the formula contributes to the final outcome, making BMI calculation straightforward for anyone to perform.

Population Data Analysis Using BMI

BMI isn’t just useful for individual analysis; it plays a crucial role in understanding population health trends. When considering a group with variation in height and weight, analyzing BMI ranges helps gauge overall health markers within that group.To explore the range of possible BMIs within a given population, consider the extremes of body metrics:

• **Max BMI scenario**: Achieved with the highest weight and shortest height. - For example, using a weight of 160 kg and a height of 1.50 m: - Calculate the square of the height: \((1.50)^2 = 2.25\). - Substitute into the formula for maximum BMI: \[ \text{Max BMI} = \frac{160}{2.25} \approx 71.1 \]
• **Min BMI scenario**: Achieved with the lowest weight and tallest height. - For example, using a weight of 45 kg and a height of 1.90 m: - Calculate the square of the height: \((1.90)^2 = 3.61\). - Substitute into the formula for minimum BMI: \[ \text{Min BMI} = \frac{45}{3.61} \approx 12.5 \]

These calculations can highlight the diversity in body composition across a population, providing insights into potential health risks or trends.