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Chapter 1: Problem 3

Your backyard has brick walls on both ends. You measure a distance of \(23.4 \mathrm{~m}\) from the inside of one wall to the inside of the other. Each wall is \(29.4 \mathrm{~cm}\) thick. How far is it from the outside of one wall to the outside of the other? Pay attention to significant figures.

Short Answer

Expert verified
The distance from the outside of one wall to the outside of the other is 24.0 m.

Step by step solution

01

Convert Wall Thickness to Meters

The thickness of each wall is given in centimeters which needs to be converted to meters as we are dealing with measurements in meters. Hence, convert 29.4 cm to meters: \(29.4\, \text{cm} = 0.294\, \text{m}\).
02

Calculate Combined Wall Thickness

Since the problem asks for the distance from the outside of one wall to the outside of the other, we need to account for the thickness of both walls. Calculate the combined thickness of the two walls: \(2 \times 0.294\, \text{m} = 0.588\, \text{m}\).
03

Add Wall Thickness to the Internal Distance

Add the combined thickness of the walls to the measured distance between the inner sides: \(23.4\, \text{m} + 0.588\, \text{m} = 23.988\, \text{m}\).
04

Adjust for Significant Figures

The original measurement, 23.4 m, has three significant figures. The added wall thickness (0.588 m) also lends itself to three significant figures for a consistent level of precision. Thus, round the calculated total distance to three significant figures: \(23.988\, \text{m} \approx 24.0\, \text{m}\).

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Key Concepts

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

Significant Figures
Significant figures are crucial in science and mathematics as they indicate the precision of a measurement. When you measure something, the significant figures include all the numbers known with certainty plus the first uncertain or estimated number. Simply put, significant figures help convey how exact a measurement is.

In our task, the key measurement is 23.4 meters, represented with three significant figures. This means the first two digits (23) are known with certainty, while the digit 4 is estimated. Likewise, when measuring thickness converted from 29.4 cm to 0.294 m, it retains its three significant figures.

Here are basic rules to identify significant figures:
  • All non-zero numbers are always significant.
  • Zeros between non-zero numbers are significant.
  • Leading zeros are not significant.
  • Trailing zeros in a number with a decimal point are significant.
After completing a calculation like addition, always round your final answer to have the same number of significant figures as the initial measurement with the fewest significant figures.
Measurement Precision
Measurement precision is how close repeated measurements are to each other. It is vital to represent the smallest possible variation in the measured value. Precision allows scientists to know how reliable a measurement is and is often conveyed by the number of significant figures.

For example, when converting wall thickness from centimeters to meters, precision is preserved by retaining significant figures. The precision here ensures even small variations like 0.294 contribute accurately to the calculation. Ensuring precision in measurements helps in maintaining accuracy in calculations and final results.

In daily applications:
  • Use measuring tools appropriate for the level of precision needed, like a ruler or a caliper.
  • Double-check conversions between units to uphold consistency.
  • Acknowledge that precision doesn't necessarily equal accuracy.
Measurement precision ensures that we do not overstate the exactness of our findings.
Addition of Measurements
The addition of measurements is a common task in scientific and everyday calculations. When you add measurements together, it's vital to consider both the units and the significant figures. Here's how it works in our exercise.

First, ensure all measurements are in the same units; if not, convert them. In our case, convert 29.4 cm to 0.294 m, as the main measurement was in meters. Next, consider the precision by using the number of significant figures each measurement contributes.

When adding these converted values:
  • Add the internal distance and the total wall thickness.
  • Ensure the result reflects the measurement with the least number of significant figures involved.
  • In situations like this one, after adding 23.4 m and 0.588 m, our result was adjusted to three significant figures.
Therefore, maintaining the units and following significant figures rules uphold the integrity of measurement addition. The final total distance was rounded to 24.0 m, adhering to the initial level of precision represented.

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Most popular questions from this chapter

In the last century, the average age of the onset of puberty for girls has decreased by several years. Urban folklore has it that this is because of hormones fed to beef cattle, but it is more likely to be because modern girls have more body fat on the average and possibly because of estrogen-mimicking chemicals in the environment from the breakdown of pesticides. \(\mathrm{A}\) hamburger from a hormone-implanted steer has about \(0.2 \mathrm{ng}\) of estrogen (about double the amount of natural beef). A serving of peas contains about 300 ng of estrogen. An adult woman produces about \(0.5 \mathrm{mg}\) of estrogen per day (note the different unit!). (a) How many hamburgers would a girl have to eat in one day to consume as much estrogen as an adult woman's daily production? (b) How many servings of peas? (answer check available at lightandmatter.com)

Assume a dog's brain is twice as great in diameter as a cat's, but each animal's brain cells are the same size and their brains are the same shape. In addition to being a far better companion and much nicer to come home to, how many times more brain cells does a dog have than a cat? The answer is not \(2 .\)

In a computer memory chip, each bit of information (a 0 or a 1 ) is stored in a single tiny circuit etched onto the surface of a silicon chip. The circuits cover the surface of the chip like lots in a housing development. A typical chip stores \(64 \mathrm{Mb}\) (megabytes) of data, where a byte is 8 bits. Estimate (a) the area of each circuit, and (b) its linear size.

You take a trip in your spaceship to another star. Setting off, you increase your speed at a constant acceleration. Once you get half-way there, you start decelerating, at the same rate, so that by the time you get there, you have slowed down to zero speed. You see the tourist attractions, and then head home by the same method. (a) Find a formula for the time, \(T\), required for the round trip, in terms of \(d\), the distance from our sun to the star, and \(a\), the magnitude of the acceleration. Note that the acceleration is not constant over the whole trip, but the trip can be broken up into constant-acceleration parts. (b) The nearest star to the Earth (other than our own sun) is Proxima Centauri, at a distance of \(d=4 \times 10^{16} \mathrm{~m}\). Suppose you use an acceleration of \(a=10 \mathrm{~m} / \mathrm{s}^{2}\), just enough to compensate for the lack of true gravity and make you feel comfortable. How long does the round trip take, in years? (c) Using the same numbers for \(d\) and \(a\), find your maximum speed. Compare this to the speed of light, which is \(3.0 \times 10^{8}\) ? Itextup \(\\{\mathrm{m}\\} \wedge\) textup\\{s\\}. (Later in this course, you will learn that there are some new things going on in physics when one gets close to the speed of light, and that it is impossible to exceed the speed of light. For now, though, just use the simpler ideas you've learned so far.) (answer check available at lightandmatter.com)

A person is parachute jumping. During the time between when she leaps out of the plane and when she opens her chute, her altitude is given by an equation of the form \(y=b-c\left(t+k e^{-t / k}\right)\) where \(e\) is the base of natural logarithms, and \(b, c\), and \(k\) are constants. Because of air resistance, her velocity does not increase at a steady rate as it would for an object falling in vacuum. (a) What units would \(b, c\), and \(k\) have to have for the equation to make sense? (b) Find the person's velocity, \(v\), as a function of time. [You will need to use the chain rule, and the fact that \(\left.d\left(e^{x}\right) / d x=e^{x} .\right]\) (answer check available at lightandmatter.com) (c) Use your answer from part (b) to get an interpretation of the constant \(c\). [Hint: \(e^{-x}\) approaches zero for large values of \(\left.x .\right]\) (d) Find the person's acceleration, \(a\), as a function of time.(answer check available at lightandmatter.com) (e) Use your answer from part (d) to show that if she waits long enough to open her chute, her acceleration will become very small.
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