Problem 40 - Calculus for Biology and Medicine

Question

Compare a square with side length \(1 \mathrm{~m}\) against a square with side length \(100 \mathrm{~m}\). The area of the larger square is order(s) of magnitude larger than the area of the smaller square.

Step-by-Step Solution

Answer

The area of the larger square is 4 orders of magnitude larger than the area of the smaller square.

Step 1: Calculate the Area of the Smaller Square

First, find the area of the smaller square, which has a side length of \(1 \mathrm{~m}\). The formula to calculate the area of a square is given by \(A = s^2\), where \(s\) is the side length of the square. Therefore, the area of the smaller square is given by: \[ A = 1^2 = 1 \mathrm{~m^2}. \]

Step 2: Calculate the Area of the Larger Square

Next, calculate the area of the larger square, which has a side length of \(100 \mathrm{~m}\). Using the same formula, \(A = s^2\), we have: \[ A = 100^2 = 10,000 \mathrm{~m^2}. \]

Step 3: Determine the Order of Magnitude Difference

To find the order(s) of magnitude by which the area of the larger square exceeds that of the smaller square, divide the area of the larger square by the area of the smaller square: \[ \text{Factor} = \frac{10,000 \mathrm{~m^2}}{1 \mathrm{~m^2}} = 10,000. \]

Step 4: Convert the Factor to Orders of Magnitude

Since each order of magnitude represents a tenfold increase, count the number of zeros in 10,000 to find how many times it increases by a factor of ten. 10,000 contains four zeros, indicating that it is \(10^4\), suggesting a four orders of magnitude difference.

Key Concepts

Understanding Area Calculation

Area calculation involves figuring out the surface size of a shape or space using a mathematical formula. For squares, this is straightforward. Suppose you have a square with side length measured in meters. The formula to find its area is:

• The formula: \[ A = s^2 \] where \( A \) is the area and \( s \) is the side length.
• For a smaller square with side \(1 \text{ m}\), this becomes \( A = 1^2 = 1 \text{ m}^2 \).
• For a larger square with side \(100 \text{ m}\), the area is \( A = 100^2 = 10,000 \text{ m}^2 \).

These calculations illustrate how an increase in the side length leads to a corresponding increase in the area. Keep in mind that the area grows drastically larger as the side length increases.

Exploring the Order of Magnitude

The concept of order of magnitude refers to the scale or size of a value, based on powers of ten. When comparing sizes or values that differ greatly, it's practical to use orders of magnitude to express how much larger or smaller one value is compared to another. Each order of magnitude indicates an increase or decrease by a factor of ten. Let's consider our exercise:

• The area of the smaller square is \( 1 \text{ m}^2 \).
• The area of the larger square is \( 10,000 \text{ m}^2 \).
• This results in a factor of \( 10,000 \div 1 = 10,000 \).
• In terms of powers of ten, \( 10,000 \) is expressed as \( 10^4 \).

This indicates that the larger square's area is four orders of magnitude greater than the smaller one's, meaning it is 10,000 times larger.

Making Mathematical Comparisons

Mathematical comparison is the process of examining two or more quantities to determine their relative sizes or values. This may involve using ratios, differences, or factors of increase. In the context of our exercise, the comparison focuses on the areas of two squares that differ significantly in size.To understand the difference:

• First, recognize that each side length increase exponentially affects the area.
• The smaller square had an area of \( 1 \text{ m}^2 \), while the larger had \( 10,000 \text{ m}^2 \).
• The comparison was facilitated by dividing the larger area by the smaller \( \left( \frac{10,000}{1} \right) \).

This reveals that the larger square holds a much greater surface area, emphasized by it being multiple magnitudes larger. Such comparisons help in illustrating how exponential growth can impact measurements of space. This understanding brings clarity to the power of exponential increases in geometry.