Exercises: Approximations and errors

Approximate the decimal number $4.8275$ to two decimal places using the truncation method and the rounding method.

Calculate the absolute error made when using the approximation $0.33$ for the exact value of the fraction $\frac{1}{3}$. Express the result as a fraction.

From the previous exercise, calculate the relative error made when approximating $\frac{1}{3}$ by $0.33$. Express the result as a percentage.

State the absolute error bound assumed when stating that the weight of an object, rounded to three decimal places, is $12.458$ lb.

A digital thermometer shows a temperature of $38.2^\circ \text{C}$. If a high-precision instrument shows that the real temperature is $38.45^\circ \text{C}$, determine the absolute error and the relative error of the digital thermometer's measurement.

Approximate the irrational number $\sqrt{7}$ to three decimal places, clearly indicating which is the lower bound approximation and which is the upper bound approximation.

Two different measurements have been taken. In the first, a relative error of $0.015$ has been made. In the second, the relative error was $1.2\%$. Determine which of the two measurements is more precise and justify your answer.

A kitchen scale gives a reading of $250$ oz with an absolute error bound of $5$ oz. Write the range of values in which the real weight of the object lies in the form of a mathematical interval.

During a track and field competition, the head judge measures the length of the track and records $399.5$ ft. If the official and exact length of the track must be $400$ ft, calculate the absolute error and the percentage relative error of the judge's measurement. Do you consider it an acceptable measurement if the regulations allow a maximum relative error of $0.2\%$?

In a chemistry laboratory, a student uses the value $6.02 \cdot 10^{23}$ for Avogadro's number in their calculations. If the most precise value currently accepted is $6.02214 \cdot 10^{23}$, determine the absolute error in scientific notation and the percentage relative error made by the student.

An architect measures the dimensions of a rectangular room, obtaining $4.5$ ft long and $3.2$ ft wide. He knows that his tape measure has an absolute error bound of $0.1$ ft for each measurement. Calculate the measured area and determine between which maximum and minimum values the real area of the room lies.

When making the budget for a construction project, the cost of materials is estimated to be $\$15\,400$. At the end of the project, the invoices add up to a real total of $\$15\,825$. Calculate the absolute and relative error of the initial estimate. If the construction company assumes the losses if the relative error exceeds $3\%$, who will pay the difference?

The size of a computer file is $4567.89$ Megabytes (MB). When copying it to a hard drive, the operating system truncates the value to one decimal place, while a compression program rounds it to one decimal place. Calculate the difference between the absolute error made by the operating system and that made by the compression program.

To calculate the density of a metal block, its mass is measured obtaining $m = 12.4$ lb and its volume obtaining $V = 0.004$ ft$^3$. If both measurements have been rounded to the last decimal digit shown, determine the absolute error bound of the mass and the volume.

The mean distance from the Earth to the Sun is approximately $149\,597\,870$ miles. A high school textbook states that the distance is $1.5 \cdot 10^8$ miles. Calculate the percentage relative error introduced by the textbook when using this approximation in scientific notation.

Two carpenters cut wooden slats that must measure exactly $\frac{5}{7}$ of a foot. The first carpenter cuts the slat to $0.71$ ft and the second cuts it to $0.72$ ft. Without using decimals, operating with fractions, determine which of the two has made a smaller absolute error.

In the manufacture of a cylindrical part for a high-precision engine, the radius is required to be $r = 2.50$ in and the height $h = 8.00$ in. If the machinery has a relative error bound of $1\%$ in any linear measurement, show algebraically whether the maximum relative error in the calculation of the volume of the cylinder ($V = \pi r^2 h$) will be $1\%$, $2\%$ or $3\%$. Assume that the relative errors are small enough to add their linear approximations.

A financial analyst estimates the price of a stock on the stock market. We know that the relative error of his estimate was exactly $4\%$. If the estimated value he published was $\$26.00$, set up and solve the necessary equations to find the two possible exact real values that the stock could have.

It is known that when approximating a number $x$ by $3.14$, the absolute error made is less than $0.005$. On the other hand, when approximating the same number $x$ by $3.146$, the absolute error is less than $0.002$. Express both conditions as intervals, calculate their intersection and determine the range of possible values for the exact number $x$.

Prove mathematically that if a positive number $x$ is rounded to the nearest whole number obtaining an approximation $A$, the relative error $E_r$ always satisfies the inequality $E_r \le \frac{0.5}{A - 0.5}$. Apply this formula to find the maximum relative error bound when rounding to $A = 5$.

The propagation speed of a signal in a fiber optic network is calculated using the formula $v = \frac{d}{t}$. A distance of $d = 1200 \pm 5$ ft and a time of $t = 0.006 \pm 0.0001$ s are measured. Calculate the maximum and minimum possible values for the speed $v$, and deduce from them the general absolute error bound of the speed measurement.

A student must calculate the value of $y = \sqrt{x}$. He knows that the exact value of $x$ belongs to the interval $[15.8, 16.2]$. If he decides to use $x = 16$ to simplify, obtaining $y = 4$, determine the upper bound of the absolute error he is making in the variable $y$.

The following diagram shows the confidence interval of a measurement in a physics experiment, where the central point represents the approximate value $V_a$ and the endpoints represent the limits given by the absolute error bound $E_a$. If the endpoints of the interval are $14.35$ and $14.65$, determine the approximate value $V_a$, the absolute error bound $E_a$ and the maximum possible relative error.

When calculating the compound interest of an initial capital of $\$10\,000$ at $5\%$ per year for $10$ years, an accountant decides to truncate the annual growth rate from $1.05$ to $1.0$ in his preliminary calculations. Calculate the absolute error and the relative error he makes in the calculation of the final capital. What conclusion can you draw about the propagation of errors in operations with high exponents?