Problem 38 - Calculus for Biology and Medicine

Question

Both the La Plata river dolphin (Pontoporia blainvillei) and the sperm whale ( Physeter macrocephalus) belong to the suborder Odontoceti (individuals that have teeth). A La Plata river dolphin weighs between 30 and \(50 \mathrm{~kg}\), whereas a sperm whale weighs between 35,000 and \(40,000 \mathrm{~kg}\). A sperm whale is order(s) of magnitude heavier than a La Plata river dolphin.

Step-by-Step Solution

Answer

The sperm whale is three orders of magnitude heavier than the La Plata river dolphin.

Step 1: Understand Orders of Magnitude

An 'order of magnitude' is a class in a system of classification determined by size, typically in powers of ten. When one entity is an order of magnitude larger than another, it means it is about ten times larger.

Step 2: Determine the Weight Range of Both Animals

Identify the weight range of the La Plata river dolphin as 30 to 50 kg and the sperm whale as 35,000 to 40,000 kg.

Step 3: Calculate the Order of Magnitude Difference

To find the order of magnitude difference, compare the powers of ten between the weights of the two animals. The dolphin weighs up to a maximum of 50 kg, represented as one to a few times 10^1, and the whale's weight is represented between 35,000 to 40,000 kg, corresponding to a few times 10^4. Subtract the exponents: 4 - 1 = 3.

Step 4: Conclude the Solution

The sperm whale is three orders of magnitude heavier than the La Plata river dolphin, as the difference in their weight corresponds to a factor of 10^3 or 1,000.

Key Concepts

Weight Comparison

In the world of biology and physics, understanding how to compare weights is a fundamental skill. When comparing two objects, such as the La Plata river dolphin and the sperm whale, we must look at their weight ranges to see how they differ.
The dolphin weighs between 30 kg and 50 kg. On the other hand, the sperm whale weighs much more, ranging from 35,000 kg to 40,000 kg.
To visualize this, imagine comparing a small bag of potatoes (the dolphin) to a large elephant (the whale).

• The dolphin's weight falls within the single tens (10s) of kilograms.
• The whale's weight soars into tens of thousands (10,000s) of kilograms.

This comparison helps highlight the substantial differences in their sizes, giving us insight into how vastly different members of the same suborder can be.

Mathematical Calculation

To truly understand how different in size these animals are, we turn to mathematical calculations. These calculations involve not just simple subtraction, but understanding the concept of orders of magnitude.
By taking the weights of the dolphin and the whale and expressing them as powers of ten, we can see clearer differences.

• The heaviest dolphin is roughly 50 kg, expressed in scientific terms as about \( 1 \times 10^1 \).
• The whale, at 35,000 to 40,000 kg, is expressed as \( 3.5 \times 10^4 \) to \( 4.0 \times 10^4 \).

Subtract the smaller exponent (1) from the larger ones (4) to see that the difference in scale is truly significant. These calculations give us precise numerical insights into the size disparity.

Powers of Ten

The concept of powers of ten is essential for simplifying complex comparisons, like those between vastly different weights. Powers of ten help us avoid cumbersome numbers and keep things tidy.
Here's how they work:

• Each 'power' uptick represents a factor increase of ten times.
• The dolphin, at a power of \( 10^1 \), is small compared to the whale's \( 10^4 \).

When comparing the exponential part, we find a difference of three (\( 4 - 1 = 3 \)). This difference shows us that the whale is 1,000 times heavier, or three orders of magnitude larger than the dolphin.
Such exponential differences, reflected in the use of scientific notation, allow us to grasp and easily articulate massive discrepancies without being bogged down by countless zeroes.