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Chapter 1: Problem 21

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?

Short Answer

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

Step by step solution

01

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:\[ 40000 \text{ miles} \times 1609.34 \text{ meters/mile} = 64373600 \text{ meters} \]
02

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:\[ \frac{64373600 \text{ meters}}{1.75 \text{ meters}} \]
03

Simplify the Expression

Perform the division to find out how many times 1.75 meters can fit into 64373600 meters:\[ \frac{64373600}{1.75} = 36784800 \]
04

Interpret the Result

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥檚 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.

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Most popular questions from this chapter

The density of metal mercury is \(13.6 \mathrm{~g} / \mathrm{cm}^{3}\). (a) What is this density as expressed in kilograms per cubic meter? (b) How many kilograms of mercury would be required to fill a 0.250 -L container?

The Roman Coliseum used to be flooded with water to re-create ancient naval battles. Assuming the circular floor be \(250 \mathrm{~m}\) in diameter and the water to have a depth of \(10 \mathrm{ft},\) (a) how many cubic meters of water are required? (b) How much mass would this water have in kilograms? (c) How much would the water weigh in pounds?

A spherical shell is formed by taking a solid sphere of radius \(20.0 \mathrm{~cm}\) and hollowing out a spherical section from the shell's interior. Assume the hollow section and the sphere itself have the same center location (that is, they are concentric). (a) If the hollow section takes up 90.0 percent of the total volume, what is its radius? (b) What is the ratio of the outer area to the inner area of the shell?

In the Tour de France, a bicyclist races up two successive (straight) hills of different slope and length. The first is \(2.00 \mathrm{~km}\) long at an angle of \(5^{\circ}\) above the horizontal. This is immediately followed by one \(3.00 \mathrm{~km}\) long at \(7^{\circ}\). (a) What will be the overall (net) angle from start to finish: (1) smaller than \(5^{\circ},\) (2) between \(5^{\circ}\) and \(7^{\circ}\), or (3) greater than \(7^{\circ} ?\) (b) Calculate the actual overall (net) angle of rise experienced by this racer from start to finish, to corroborate your reasoning in part (a).

The outside dimensions of a cylindrical soda can are reported as \(12.559 \mathrm{~cm}\) for the diameter and \(5.62 \mathrm{~cm}\) for the height. (a) How many significant figures will the total outside area have: (1) two, (2) three, (3) four, or (4) five? Why? (b) What is the total outside surface area of the can in square centimeters?
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