Problem 21 - College Physics

Question

If the capillaries of an average adult were unwound and spread out end to end, they would extend to a length over \(40000 \mathrm{mi}\) (Fig. 1.9). If you are \(1.75 \mathrm{~m}\) tall, how many times your height would the capillary length equal?

Step-by-Step Solution

Answer

The capillary length is 36,784,800 times your height.

Step 1: Convert Miles to Meters

First, we need to convert the length from miles to meters. We know that 1 mile is approximately 1609.34 meters. Thus, we multiply:\[ 40000 \text{ miles} \times 1609.34 \text{ meters/mile} = 64373600 \text{ meters} \]

Step 2: Calculate the Ratio

Now that we have the capillary length in meters, we need to determine how many times your height (in meters) fits into this length. Your height is given as 1.75 meters. Thus, the ratio is calculated by dividing the total length by your height:\[ \frac{64373600 \text{ meters}}{1.75 \text{ meters}} \]

Step 3: Simplify the Expression

Perform the division to find out how many times 1.75 meters can fit into 64373600 meters:\[ \frac{64373600}{1.75} = 36784800 \]

Step 4: Interpret the Result

The calculation reveals that the capillary's total length is around 36,784,800 times your height.

Key Concepts

Unit Conversion

Unit conversion is a fundamental concept in physics and mathematics, especially when dealing with problems that involve different measurement systems. In this exercise, we start with a measurement given in miles and need to convert it to meters because the subsequent calculations involve height measured in meters.
To convert from miles to meters, it is essential to know the conversion factor: 1 mile is approximately 1609.34 meters.

• This means every mile is equal to 1609.34 meters.
• Therefore, multiplying the total miles by this conversion factor gives us the length in meters.

The conversion ensures that we're working within the same unit system, resulting in correct and meaningful calculations. Understanding and applying the right conversion factors is crucial in achieving accurate results in physics problems.

Ratios and Proportions

Ratios and proportions are key mathematical tools used to compare quantities. In our problem, once we have the capillary length in meters, we employ the concept of ratios to determine how this length compares to a specific height.
Here’s how it works:

• A "ratio" quantifies the relationship between two numbers.
• By dividing the total length by a person's height, we calculate the ratio that shows how many times one can fit into the other.

In this scenario, the ratio calculated helps us understand the sheer scale of the capillary length in contrast to a person's height. Ratios can be simplified, kept in their exact form, or expressed as decimals or percentages, depending on the context or requirement of the problem.

Mathematical Calculations

Mathematical calculations are at the heart of solving physics problems, requiring precision and a good understanding of fundamental operations. After converting units and setting up the ratios, the next step involves actually performing the calculations.
Here's what we do:

• We divide the total capillary length in meters by the person's height in meters.
• This simple division provides us with a specific numeric value indicating how many times the height fits into the total length.

Executing calculations accurately is vital to deriving meaningful conclusions. It ensures that subsequent interpretations and decisions based on these numbers are reliable and valid. Always double-check calculations to minimize errors, particularly when dealing with large or small numbers.

Length Measurement

Length measurement is a basic but essential concept often encountered in physics and everyday applications. Understanding length involves not just the numbers but also grasping the systems of measurements and their significance.
In this problem:

• The initial length measurement provided in miles shows the vast expanse that the capillaries could create if laid out straight.
• Measuring multiple facets of everyday life, like height in meters, helps provide context and relativity to such large numbers.

Length serves a vital role not only in problems like these but also in fields from architecture to clothing design. Accurately interpreting and calculating lengths is fundamental for science and technology advancements.