Problem 37 - Calculus for Biology and Medicine

Question

The longest known species of worms is the earthworm Microchaetus rappi of South Africa; in 1937, a 6.7-m-long specimen was collected from the Transvaal. The shortest worm is Chaetogaster annandalei, which measures less than \(0.51 \mathrm{~mm}\) in length. \(M .\) rappi is order(s) of magnitude longer than C. annandalei.

Step-by-Step Solution

Answer

M. rappi is about 4 orders of magnitude longer than C. annandalei.

Step 1: Understanding Order of Magnitude

The term 'order of magnitude' refers to a way of describing the size or scale of a quantity using powers of ten. When comparing two numbers, if one is ten times larger than the other, it is one order of magnitude larger. If it is 100 times larger, it is two orders of magnitude larger, and so on.

Step 2: Convert Lengths to the Same Unit

To compare the orders of magnitude of the worms, we need the lengths in the same unit. Microchaetus rappi's length is already given in meters, but Chaetogaster annandalei's length is given in millimeters. Convert C. annandalei's length to meters: \[ 0.51 \text{ mm} = 0.00051 \text{ m} \]

Step 3: Calculate the Ratio of Lengths

Now that both lengths are in meters, calculate the ratio of the length of M. rappi to C. annandalei:\[ \text{Ratio} = \frac{6.7 \text{ m}}{0.00051 \text{ m}} \]

Step 4: Simplify the Ratio to Orders of Magnitude

Perform the division to find:\[ \frac{6.7}{0.00051} \approx 13137.25 \]Next, express this number as a power of ten (in terms of orders of magnitude):\[ 13137 \approx 10^4 \]So, M. rappi is more than four orders of magnitude longer than C. annandalei.

Key Concepts

Powers of Ten

Understanding powers of ten is crucial in scientific contexts where large and small numbers are compared. This concept simplifies complex calculations by using exponents. A power of ten means multiplying 10 by itself a certain number of times.

• For example, \(10^1\) equals 10, \(10^2\) equals 100, and \(10^3\) equals 1,000.
• To express a number in powers of ten, count how many times 10 is multiplied to reach or approach that number.

Using powers of ten, we can efficiently describe large differences. For instance, the concept of 'order of magnitude' describes a number relative to another using powers of ten. If a number is 10 times larger than another, it is one order of magnitude larger. Understanding this helps in comparing vast differences, such as between worm species, where one might be significantly longer than another.

Length Conversion

When comparing measurements, a common unit of measure allows for easy evaluation. Converting between units is a key skill here. To ensure consistency, we often convert lengths into the same unit.

• In the worm example, we have lengths given in meters and millimeters.
• Since 1 meter equals 1,000 millimeters, we can convert millimeters to meters by dividing by 1,000.
• For Chaetogaster annandalei, the conversion is: \(0.51 \text{ mm} = 0.00051 \text{ m}\).

By converting all values to meters, comparing different lengths becomes straightforward. This step is essential for subsequent calculations, such as determining ratios or orders of magnitude.

Ratio Calculation

Calculating ratios provides insights into how quantities relate to each other. It measures how many times one number contains another.

• This is a potent tool in comparing vastly different sizes or amounts.
• To find the ratio of lengths between two worms, divide the length of one worm by the length of the other.

In our example, the ratio of the lengths of Microchaetus rappi and Chaetogaster annandalei is:\[\text{Ratio} = \frac{6.7 \text{ m}}{0.00051 \text{ m}} \approx 13137.25\]This ratio shows that M. rappi is about 13,137 times longer than C. annandalei. A ratio expressed as a power of ten helps easily understand orders of magnitude, highlighting the significant difference between the two species' sizes. Here, we'd say it's over four orders of magnitude different, since \(13137 \approx 10^4\), indicating a substantial size disparity.