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Chapter 2: Problem 21

The following expression has occurred in a problem solution: $$R=\frac{(0.6700)(264,980)(6)\left(5.386 \times 10^{4}\right)}{(3.14159)\left(0.479 \times 10^{7}\right)}$$ The factor 6 is a pure integer. Estimate the value of \(R\) without using a calculator, following the procedure outlined in Section 2.5b. Then calculate \(R\), expressing your answer in both scientific and decimal notation and making sure it has the correct number of significant figures.

Short Answer

Expert verified
The estimated value of \(R\) without using a calculator is \(1.4 x 10^{5}\), and the more precise calculated value of \(R\) is \(2.0 x 10^{5}\) (in scientific notation) or \(200,000\) (in decimal notation).

Step by step solution

01

Estimating the Value

Begin by rounding all the numerical factors into one significant figure except pure integers. Hence, \n\(0.6700\) becomes \(0.7\), \n\(264,980\) becomes \(300,000\) (only one significant figure), \n\(5.386 脳 10^{4}\) becomes \(5 脳 10^{4}\), and \n\(0.479 脳 10^{7}\) becomes \(5 脳 10^{7}\). \nThe integer \(6\) remains the same. Now, replace the original numbers in \(R\) by these estimated values. The fraction becomes \((0.7)(300,000)(6)(5 x 10^{4}) / ((3)(5 x 10^{7}))\).
02

Simplify the Fraction

Next step is to simplify the fraction. We can cancel out equal terms in the numerator and denominator and rewrite \(300,000\) as \(3 x 10^{5}\) . After this simplification, the fraction becomes \((0.7)(6)(3 x 10^{5}) / (3)\).
03

Perform Calculations

The fraction now becomes \(1.4 x 10^{5}\). After performing these calculations, we find that the estimated value of \(R\) is \(1.4 x 10^{5}\).
04

Calculating Exact Value

Now, calculate the exact value using the original numbers in \(R\) given in the problem. \n\(R=(0.6700)(264,980)(6)(5.386 x 10^{4}) / ((3.14159)(0.479 x 10^{7})\). \nAfter calculating more precisely, \(R\) becomes \(1.997 x 10^{5}\), in scientific notation.
05

Final Output

The final output would be correctly rounded to the number of least significant figures from the given data, which is 2. Therefore, the final value of \(R\) is \(2.0 x 10^{5}\) in scientific notation and \(200,000\) in decimal notation.

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Key Concepts

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

Understanding Scientific Notation
Scientific notation is a way to express very large or very small numbers in a concise and consistent manner. This method helps simplify calculations and reduces the risk of errors in lengthy or complex expressions. The format involves writing numbers as a product of a coefficient and a power of ten, like this: \( a \times 10^n \)where \( a \) is a number between 1 and 10, and \( n \) is an integer.For example:
  • The number 5,386 can be written as \( 5.386 \times 10^3 \), and
  • The number 0.00479 can be expressed as \( 4.79 \times 10^{-3} \).
In the given exercise, we notice scientific notation when numbers like \( 5.386 \times 10^{4} \) are used. This helps efficiently handle the magnitude of numbers, making it easier to manage multiplication or division in mathematical expressions.
When simplifying or estimating expressions using scientific notation, it becomes easy to approximate calculations without a calculator, significantly aiding problem-solving tasks.
Mastering Estimation Techniques
Estimation techniques are essential tools for quickly approximating values and making decisions when exact numbers are not required. These techniques involve rounding numbers to the nearest significant figure to simplify calculations, allowing for quick mental math.In our exercise, we see that each number is rounded to one significant figure for estimation purposes:
  • \( 0.6700 \) is rounded to \( 0.7 \)
  • \( 264,980 \) becomes \( 300,000 \) (or \( 3 \times 10^5 \))
  • And \( 5.386 \times 10^4 \) simplifies to \( 5 \times 10^4 \)
  • \( 0.479 \times 10^7 \) turns into \( 5 \times 10^7 \)
Using estimation techniques, complex expressions like fractions can be substantially simplified, making them much easier to compute intuitively.
Rounding off to the nearest significant figures can sometimes change the result slightly, but it is often adequate for quickly understanding an expression's magnitude.
Simplifying Fractions
Fraction simplification is about reducing a fraction to its simplest form so that the numbers are as small and manageable as possible. This involves canceling out common factors in the numerator and the denominator and rearranging the expression for clarity.To illustrate this, consider the fraction from our exercise:\[\frac{(0.7)(300,000)(6)(5 \times 10^4)}{(3)(5 \times 10^7)}\]Here, \( (5 \times 10^4) \) and \( (5 \times 10^7) \) were part of larger numbers that had factors in common. By simplifying \( 300,000 \) to \( 3 \times 10^5 \) and canceling equal terms, the expression was reduced:
  • Cancel \( (5 \times 10^4) \) with \( (5 \times 10^7) \) by reducing their powers of ten,
  • Similarly, the constant numerical factors allow us to strike out \( 3 \) from the denominator since \( 3 \) is a factor of \( 3 \times 10^5 \) in the numerator.
After simplification, this results in much more manageable numbers, making it easier to compute either mentally or with minimal tools.
Mastering fraction simplification can greatly aid in estimating and solving problems, leading to more efficient and straightforward calculations.

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Most popular questions from this chapter

The following \((x, y)\) data are recorded: $$\begin{array}{|c|c|c|c|}\hline x & 0.5 & 1.4 & 84 \\\\\hline y & 2.20 & 4.30 & 6.15 \\\\\hline \end{array}$$ (a) Plot the data on logarithmic axes. (b) Determine the coefficients of a power law expression \(y=a x^{b}\) using the method of least squares. (Remember what you are really plotting \(-\) there is no way to avoid taking logarithms of the data point coordinates in this case.) (c) Draw your calculated line on the same plot as the data.

A frustrated professor once claimed that if all the reports she had graded in her career were stacked on top of one another, they would reach from the Earth to the moon. Assume that an average report is the thickness of about 10 sheets of printer paper and use a single dimensional equation to estimate the number of reports the professor would have had to grade for her claim to be valid.

A hygrometer, which measures the amount of moisture in a gas stream, is to be calibrated using the apparatus shown here: Steam and dry air are fed at known flow rates and mixed to form a gas stream with a known water content, and the hygrometer reading is recorded; the flow rate of either the water or the air is changed to produce a stream with a different water content and the new reading is recorded, and so on. The following data are taken: $$\begin{array}{cc}\hline \begin{array}{c}\text { Mass Fraction } \\\\\text { of Water, } y\end{array} & \begin{array}{c}\text { Hygrometer } \\\\\text { Reading, } R\end{array} \\\\\hline 0.011 & 5 \\\0.044 & 20 \\\0.083 & 40 \\\0.126 & 60 \\\0.170 & 80 \\ \hline\end{array}$$ (a) Draw a calibration curve and determine an equation for \(y(R)\). (b) Suppose a sample of a stack gas is inserted in the sample chamber of the hygrometer and a reading of \(R=43\) is obtained. If the mass flow rate of the stack gas is \(1200 \mathrm{kg} / \mathrm{h}\), what is the mass flow rate of water vapor in the gas?

A waste treatment pond is \(50 \mathrm{m}\) long and \(25 \mathrm{m}\) wide, and has an average depth of \(2 \mathrm{m}\). The density of the waste is \(75.3 \mathrm{lb}_{\mathrm{m}} / \mathrm{ft}^{3}\). Calculate the weight of the pond contents in \(\mathrm{lb}_{\mathrm{f}},\) using a single dimensional equation for your calculation.

State what you would plot to get a straight line if experimental ( \(x, y\) ) data are to be correlated by the following relations, and what the slopes and intercepts would be in terms of the relation parameters. If you could equally well use two different kinds of plots (e.g., rectangular or semilog), state what you would plot in each case. [The solution to Part (a) is given as an example.] (a) \(y^{2}=a e^{-b / x}\). (b) \(y^{2}=m x^{3}-n\) (c) \(1 / \ln (y-3)=(1+a \sqrt{x}) / b\) (d) \((y+1)^{2}=\left[a(x-3)^{3}\right]^{-1}\) (e) \(y=\exp (a \sqrt{x}+b)\) (f) \(x y=10^{\left[a\left(x^{2}+y^{2}\right)+b\right]}\) (g) \(y=[a x+b / x]^{-1}\)
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