Problem 110 - Calculus for Biology and Medicine

Question

Yang et al. (2014) measured the time that animals of different sizes spend urinating. For animals larger than \(1 \mathrm{~kg}\), the time spent urinating increases (slowly) with animal size. The smallest animal in their study was a cat (mass \(5 \mathrm{~kg}\), duration of urination 18s) and the largest was an elephant (mass \(5000 \mathrm{~kg}\), duration of urination \(29 \mathrm{~s}\) ). Make a sketch of time spent urinating as a function of animal size.

Step-by-Step Solution

Answer

Plot two points for cat and elephant, then draw a slow increasing trend line.

Step 1: Identify Variables

Identify the variables involved: Mass of the animal (in kg) and Duration of urination (in seconds).

Step 2: Identify Data Points

We have two key data points. Cat: Mass = 5 kg, Duration = 18 s, and Elephant: Mass = 5000 kg, Duration = 29 s.

Step 3: Set Up Axes on Graph

On a graph, set the x-axis as the mass of the animal in kg and the y-axis as the duration of urination in seconds. Choose an appropriate scale that fits both data points.

Step 4: Plot Data Points

Plot the two data points identified earlier on the graph: (5 kg, 18 s) for the cat and (5000 kg, 29 s) for the elephant.

Step 5: Draw the Trend Line

Since the duration of urination increases slowly with animal size, draw a line connecting the two plotted points, showing a slow increase or a slightly upward curve.

Key Concepts

Variables Identification

In the context of this exercise, identifying variables is a fundamental step in understanding the relationship between the size of an animal and the time it spends urinating. The objective here is to recognize and define the two primary variables that are changing and being measured. We have:

• **Mass of the Animal (Independent Variable):** Measured in kilograms (kg), this variable represents the size or weight of the animal.
• **Duration of Urination (Dependent Variable):** Measured in seconds (s), it signifies how long the animal takes to urinate.

The mass of the animal serves as the independent variable since it influences how long the urination lasts, which is our dependent variable. This relationship helps illustrate causation, as changes in the mass provide insights into variations in the urination duration. By identifying these variables, we can reliably proceed to visualize the data and analyze the trends.

Data Visualization

Data visualization is a powerful tool that helps us represent complex data in a simplified manner through graphical representations like charts or graphs. In this exercise, visualizing the relationship between animal size and urination duration involves plotting the identified variables on a graph.

**Setting Up the Graph**

• **X-Axis (Horizontal):** Represents the mass of the animal.
• **Y-Axis (Vertical):** Represents the duration of urination.

Once the axes are set, we need to ensure that the scales are appropriate to capture both the smallest animal (cat) and the largest animal (elephant). With the x-axis ranging from 5 kg to 5000 kg and the y-axis ranging from 18 seconds to 29 seconds, you should mark these axes accordingly.

**Plotting the Data Points**

• **Cat:** Plot at (5 kg, 18 s)
• **Elephant:** Plot at (5000 kg, 29 s)

Bring these points onto your graph as distinct marks for clarity and accuracy. A clear data visualization can bridge the gap between raw data and understandable insights.

Trend Analysis

Trend analysis involves examining data points to identify any patterns or general directions. In our exercise, we're focusing on understanding how the duration of urination generally changes with the size of the animal.

Given our two data points: (5 kg, 18s) for the cat and (5000 kg, 29s) for the elephant, it’s evident that as the animal size increases, the urination duration increases slightly too.

**Drawing the Trend Line**

• **Connect Data Points:** Draw a line between the two plotted points.
• **Interpret the Slope:** The line's gentle upward slope indicates a slow, yet positive correlation between mass and urination duration.

This analysis helps us infer that larger animals might require more time to urinate, but the increase in time does not appear drastic. By examining these trends, researchers can propose theories regarding anatomical or physiological reasons for this behavior in different-sized animals.