Exercises: Approximations and errors

Approximate the decimal number $4,8275$ to two decimal places (hundredths) using truncation and rounding.

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.

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 asserting that the weight of an object, rounded to three decimal places (thousandths), is $12,458$ kg.

A digital thermometer displays a temperature of $38,2^\circ \text{C}$. If a high-precision instrument shows that the actual temperature is $38,45^\circ \text{C}$, determine the absolute error and the relative error of the digital thermometer reading.

Approximate the irrational number $\sqrt{7}$ to three decimal places (thousandths), stating clearly which is the lower estimate (approximation by defect) and which is the upper estimate (approximation by excess).

Two different measurements have been carried out. In the first, a relative error of $0,015$ was 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 $250$ g with an absolute error bound of $5$ g. Write in the form of a mathematical interval the range of values in which the actual weight of the object lies.

In a 4th-year ESO class, Mateo and his classmate Alba are comparing the quality of two measurements taken in the Science lab:

• Measurement 1: The thickness of a 1-euro coin, whose true value is $2,33$ mm, is measured to be $2,5$ mm.

• Measurement 2: The length of the lab bench, whose true value is $180,0$ cm, is measured to be $181,0$ cm.

Mateo states: "The measurement of the bench is clearly worse because we missed by $1$ cm ($10$ mm), whereas for the coin we only deviated by $0,17$ mm; therefore, the measurement of the coin has much greater precision and smaller error."

Identify Mateo's conceptual error, calculate the absolute error and the percentage relative error for each measurement, and write a reasoned explanation refuting his conclusion and indicating which of the two measurements is actually of higher quality.

To prepare an end-of-year snack party, you go to the supermarket with a maximum budget of $25$ € and note down the exact prices of four items: soft drinks ($7,89$ €), snacks ($6,45$ €), savoury pastries ($5,90$ €), and a cake ($4,85$ €).

1. a) Before adding with decimals, make a quick estimate of the total cost by rounding the price of each item to the nearest euro. According to your preliminary estimate, is your $25$ € budget enough?
2. b) Calculate the exact actual cost and determine the absolute error and percentage relative error made with your estimate.
3. c) In light of the actual result, assess whether your estimate was an appropriate strategy to know if you could pay at the checkout. Justify why this situation occurred by analyzing the direction of the roundings you performed.

During an athletics competition, the head judge measures the length of the track and records $399,5$ m. If the official and exact length of the track should be $400$ m, calculate the absolute error and the percentage relative error of the judge's measurement. Do you consider this measurement acceptable 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 accurate currently accepted value is $6,02214 \cdot 10^{23}$, determine the absolute error in standard form and the percentage relative error made by the student.

An architect measures the dimensions of a rectangular room, obtaining a length of $4,5$ m and a width of $3,2$ m. He knows that his tape measure has an absolute error bound of $0,1$ m for each measurement. Calculate the measured area and determine between which maximum and minimum values the actual area of the room lies.

When preparing the budget for a construction project, the cost of materials is estimated to be $15\,400$ €. Upon completion of the project, the invoices sum to an actual total of $15\,825$ €. Calculate the absolute and relative errors of the initial estimate. If the construction company covers losses whenever 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, whereas 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 as $m = 12,4$ kg and its volume as $V = 0,004$ m$^3$. If both measurements have been rounded to the last decimal place shown, determine the absolute error bound for the mass and for the volume.

The average distance from the Earth to the Sun is approximately $149\,597\,870$ km. A secondary school textbook states that the distance is $1,5 \cdot 10^8$ km. Calculate the percentage relative error introduced by the textbook when using this approximation in standard form.

Two carpenters cut wooden battens that must measure exactly $\frac{5}{7}$ of a metre. The first carpenter cuts the batten to $0,71$ m and the second cuts it to $0,72$ m. Without using decimals, working with fractions, determine which of the two has made a smaller absolute error.

The automated weighing system of a parcel delivery company records the weight of a package as $4,6$ kg after rounding to the nearest tenth of a kilogram.

1. a) Write the continuous interval $[a;\ b)$ in which the actual weight $P$ of the package lies and determine the maximum bound on the absolute error guaranteed by this rounding.
2. b) Construct and justify an example of an actual value for $P$ that simultaneously satisfies two conditions: that when rounded to the nearest tenth it results in $4,6$ kg, but if it had been truncated to the tenths it would have resulted in $4,5$ kg.
3. c) For the value you proposed in the previous section, calculate the absolute error and the percentage relative error of both approximations (rounding and truncation) and explain in a reasoned manner which of the two methods provides a better approximation in this case.

An agricultural cooperative in Jaén needs to invoice an order of $1200$ bottles of extra virgin olive oil. The technical and commercial data are:

• Capacity of each bottle: $0,75$ litres.

• Production cost: $6,837$ € per litre.

• Applicable commercial discount on the total cost: $8,5\%$ (that is, $91,5\%$ is paid).

Two administrative clerks propose two different calculation methods:

• Method 1 (Mario): Calculates the cost of one bottle ($0,75 \cdot 6,837$) and rounds the result to the nearest euro cent; applies the $8,5\%$ discount to that amount and rounds again to the nearest cent; finally, multiplies this unit price by the $1200$ bottles.

• Method 2 (Sonia): Performs a single combined operation without intermediate roundings and rounds only the final invoice figure to the nearest euro cent.

1. a) Perform the calculations step by step with both methods and obtain the final amount in euros for each one.
2. b) Taking the unrounded result of Method 2 as the exact value, calculate the absolute error and the percentage relative error that Mario's method introduces into the invoice total.
3. c) Critically justify which of the two methods should be chosen in economic and scientific management, explaining the effect of rounding in intermediate steps.

In the manufacturing of a cylindrical part for a high-precision engine, the radius is required to be $r = 2,50$ cm and the height $h = 8,00$ cm. If the machinery has a relative error bound of $1\%$ for any linear measurement, prove algebraically whether the maximum relative error in calculating 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 sum their linear approximations.

A sports tracking app estimates the distance of a cycle route. We know that the relative error of its estimate was exactly $4\%$. If the estimated value displayed by the app is $26,00$ km, set up and solve the necessary equations to find the two possible exact real values that the distance 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 upper bound for the relative error when rounding to $A = 5$.

The speed of propagation of a signal in a fibre optic network is calculated using the formula $v = \frac{d}{t}$. A distance of $d = 1200 \pm 5$ m 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 overall absolute error bound of the speed measurement.

A student must calculate the value of $y = \sqrt{x}$. They know that the exact value of $x$ belongs to the interval $[15,8; 16,2]$. If they decide to use $x = 16$ to simplify, obtaining $y = 4$, determine the upper bound of the absolute error being made 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 compound interest on an initial capital of $10\,000$ € at $5\%$ per annum 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 made in the final capital calculation. What conclusion can you draw about the propagation of errors in operations with large exponents?

The State Meteorological Agency (AEMET) and the Ebro Hydrographic Confederation are studying the volume of water collected in a reservoir after a storm. The surface of the reservoir is modeled in a simplified way as a rectangle whose satellite dimensions are $L = 3,4$ km long and $A = 1,2$ km wide, both measurements rounded to the nearest tenth of a kilometre. The rain gauge station recorded a precipitation of $h = 45$ mm of rain, rounded to the nearest millimetre.

1. a) Determine the absolute error bound for each of the three measurements ($L$, $A$, and $h$) and calculate the upper bound of the percentage relative error for each of them.
2. b) The total volume of fallen water is given by $V = L \cdot A \cdot h$. Applying the property of propagation of relative errors for products, deduce the bound on the total percentage relative error affecting the calculation of $V$.
3. c) Calculate the estimated volume $V$ in cubic metres ($m^3$) and, using the relative error bound from the previous section, calculate the margin of uncertainty in cubic metres (absolute error bound of $V$) and the interval between whose extreme values the actual volume of precipitated water safely lies.

An aeronautical company located in Seville manufactures cylindrical titanium sensors. The mass of each sensor is calculated with the formula $M = d \cdot \pi \cdot r^2 \cdot h$, where the density $d$ is known exactly. Due to strict aviation safety requirements, the relative error committed in determining the mass $M$ must not exceed $1,2\%$.

The machining process already guarantees that the height $h$ is measured with a relative error of $0,4\%$.

1. a) Applying the properties of relative error propagation in products and powers, set up an inequality and deduce the maximum percentage relative error that can be tolerated in the measurement of the radius $r$.
2. b) If the nominal radius of the part is $r = 15,0$ mm, determine the maximum bound on the absolute error allowed for the measurement of $r$.
3. c) In the workshop, two digital callipers are available to measure the radius: calliper A rounds to the nearest tenth of a millimetre and calliper B rounds to the nearest hundredth of a millimetre. Analyze in a reasoned manner what absolute error bound each instrument introduces and justify whether calliper A is sufficient to comply with the regulations or if it is essential to use calliper B.