Exercises: Approximation and errors

A Formula 1 driver has recorded a lap time of $1.24572$ minutes. To simplify the graphics in the television broadcast, approximations of this time are required.

Calculate:

1. a) The time truncated to the thousandths.

1. b) The time rounded to the thousandths.

1. c) The time rounded to the hundredths.

1. d) Briefly explain why the result of truncating and the result of rounding to the thousandths are different in this particular case.

In an electronics shop, the actual price of a smartphone is $456.28$ €. However, on the shop window advertising sign, the owner decides to display the price rounded to the nearest unit to make it more visually appealing.

Calculate:

• The approximate value displayed on the sign.

• The absolute error made with this approximation.

• The relative error expressed as a percentage (to two decimal places).

When measuring the thickness of a textbook with a ruler whose smallest division is millimetres, we obtain a value of $2.4$ cm.

State:

• The absolute error bound of this measurement.

• Between which values the actual thickness of the book lies, expressing the result as an interval.

We have two different measurements made by different operators:

1. A carpenter measures a wooden slat of $2.5$ metres and makes an absolute error of $1$ cm.

2. A surveyor measures a plot of land of $450$ metres and makes an absolute error of $1$ metre.

Calculate the relative error of each measurement and give reasons to determine which of the two approximations is of better quality (more precise).

The golden ratio, represented by the Greek letter $\phi$, has an exact value of $\frac{1 + \sqrt{5}}{2}$. A student uses a calculator and finds its value to be approximately $1.6180339...$

If the student decides to take $1.62$ as an approximation of the golden ratio:

• Is it a rounding or a truncation to hundredths?

• Calculate the absolute error made (take the calculator value as the true value).

• Is this absolute error less than one thousandth?

Identify the number of significant figures in the following data obtained in a chemistry laboratory:

• $0.00405$ grams of a reactive compound.

• $120.70$ millilitres of distilled water.

• $3.50 \times 10^4$ particles per cubic centimetre.

• $2000$ bacteria in a sample (rounded to the nearest unit).

The mass of the Earth is approximately $5,972,200,000,000,000,000,000,000$ kg. In science textbooks, this number is usually expressed in scientific notation rounded to two decimal places (three significant figures).

1. a) Write the mass of the Earth in scientific notation with all its digits.

1. b) Write the rounded approximation that would be used in a textbook.

1. c) Calculate the absolute error bound associated with that rounding.

To bake a giant cake, a baker adds the masses of the three main ingredients, which he has weighed on an industrial scale:

• Flour: $12.4$ kg (rounded to tenths).

• Sugar: $5.75$ kg (rounded to hundredths).

• Butter: $2.125$ kg (rounded to thousandths).

Calculate the total sum of the ingredients and round the final result so that it is consistent with the least precise measurement of the three. Justify your choice.

For an Economics project in 5th Year, Aoife and Seán are analyzing the budget variances of two items:

• Item A (computer equipment): actual cost of $820$ €, budgeted at $800$ €.

• Item B (stationery supplies): actual cost of $12$ €, budgeted at $10$ €.

Seán claims: "In item B we were only off by $2$ €, while in item A we were off by $20$ €. Since $2 < 20$, the stationery estimate was ten times more accurate than the computer equipment one".

1. a) Identify the conceptual error in Seán's statement and explain why absolute error is not a valid indicator for comparing the quality of these two approximations.
2. b) Calculate the absolute error and the percentage relative error made in each of the two items.
3. c) Based on the results, determine with mathematical rigor which of the two budget estimates was of better quality.

A university student spends exactly $1.85$ € every school day on a coffee at the campus cafeteria. In a semester there are exactly $78$ school days.

1. a) Make a quick mental estimate of the total expenditure on coffee for the semester: first round the unit price to the nearest euro and the number of days to the nearest ten. Write down your estimate.
2. b) Calculate the exact actual expenditure and determine the absolute error and the percentage relative error you made with your initial estimate.
3. c) Reflect and reason without doing new calculations: was it foreseeable that your estimate would overestimate the actual expenditure rather than underestimate it? Justify it by analyzing in which direction each of the two factors was rounded.

Two sound technicians are calibrating equipment for a music festival. The first technician measures the length of an antenna cable and obtains $15.25$ m, when its actual value is $15.30$ m. The second technician measures the latency of a digital signal and obtains $0.018$ s, the actual value being $0.020$ s.

1. a) Calculate the absolute error made by each technician.
2. b) Determine which of the two measurements is more precise by comparing their relative errors.
3. c) If the first technician rounds their measurement to one decimal place, what would be the new relative error made?

An electronic component manufacturing company produces resistors that must have a nominal value of $470$ ohms ($\Omega$). Quality control permits a margin of error (tolerance) such that the actual value $R$ must satisfy the condition $|R - 470| \leq 15$.

1. a) Express the permitted resistance values as an interval on the real number line.
2. b) If a measured resistor gives a value of $482.5 \Omega$, is it within the tolerance margin?
3. c) Calculate the upper bound on the maximum relative error accepted by the company for this component.

The radius of a circular skating rink has been measured with a tape measure, obtaining a value of $12.4$ metres. It is known that the absolute error made in this measurement is, at most, $0.05$ metres.

1. a) Write the interval in which the actual value of the radius lies.
2. b) Calculate the maximum and minimum possible values for the area of the rink (use $\pi \approx 3.14159$).
3. c) If we take as an approximation of the area the value calculated with the radius of $12.4$ m, what is the upper bound on the absolute error for that area?

In a biology laboratory, the number of bacteria on a culture plate is estimated to be $4.25 \times 10^7$, with a relative error of $2\%$.

1. a) Calculate the absolute error made in the estimation.
2. b) Between which two values does the actual number of bacteria lie?
3. c) If after one hour the number of bacteria triples, express the new total in standard form and calculate its order of magnitude.

Consider the real numbers $A = \sqrt{7}$ and $B = \frac{22}{8}$.

1. a) Represent the number $A$ exactly on the number line using Pythagoras' theorem.
2. b) Approximate both numbers by rounding to three decimal places.
3. c) What relative error is made when approximating the sum $A + B$ by the number $5.4$?

An investor analyses the profitability of a cryptocurrency. The current value is $0.0004582$ € per unit. To simplify calculations on their mobile app, they decide to truncate this value to the hundred-thousandths (fifth decimal place).

1. a) Write the truncated value in standard form.
2. b) Calculate the absolute error and the relative error made by this truncation.
3. c) What would the relative error be if, instead of truncating, they had rounded to the same digit?

Given the following sets of real numbers:

$A = \{x \in \mathbb{R} : |x - 1| < 3\}$

$B = [-2, 5]$

1. a) Express set $A$ as an interval.
2. b) Calculate and represent on the number line the intersection $A \cap B$ and the union $A \cup B$.
3. c) Determine whether the irrational number $\sqrt{10}$ belongs to any of these sets. Justify your answer by approximating the radical to two decimal places.

You can do it on paper and send a photo with your answer.

A seismograph scale uses logarithms to measure the energy released by an earthquake. If the energy $E$ (in joules) is related to the magnitude $M$ by the formula $\log E = 4.8 + 1.5M$:

1. a) Calculate the energy released by an earthquake of magnitude $M = 6.2$. Express the result in standard form, rounding the mantissa to two decimal places.
2. b) If we make an error of $0.1$ when measuring the magnitude ($M = 6.3$ instead of $6.2$), what is the percentage relative error made in calculating the energy?
3. c) Explain why a small error in the magnitude $M$ causes such a large error in the energy $E$.

According to data from CSO demographic publications, an Irish municipality has a registered population of $48\,600$ inhabitants, a figure that is published rounded to the nearest hundred. On the other hand, the County Council projects an average social spending of $125$ € per inhabitant, a figure rounded to the nearest euro. The County Council projects a total budget item of $G = 48\,600 \times 125 = 6\,075\,000$ €.

1. a) Write the uncertainty intervals within which the actual number of inhabitants ($P$) and the actual average spending per inhabitant ($g$) lie.
2. b) Using the relative error bound propagation formula for the product of two quantities, calculate an upper bound for the percentage relative error of the total budget item $G$.
3. c) From said relative bound, calculate the upper bound of the absolute error of the total expenditure and determine the interval of values within which the actual expenditure will definitely lie.

In an economic trends report on the year-on-year CPI of an Irish region, it is published that the inflation rate in the transport sector was $3.4$ %, a value obtained after rounding to the tenths of a percentage point.

1. a) Determine the interval of numerical values in which the actual rate $\tau$ lies.
2. b) Find the maximum bound of the absolute error and calculate the largest percentage relative error that could have been made when publishing the figure $3.4$ % compared to the unknown actual value.
3. c) If the CSO requires by quality protocol that the relative error made when publishing the rate does not exceed $0.5$ % under any circumstances, is having rounded to the tenths sufficient? If not, determine how many decimal places should have been kept at a minimum in the publication to ensure that this standard is met.

A high-precision microchip has a rectangular base whose dimensions have been measured with a laser calliper. The length obtained is $l = 4.25$ mm and the width is $w = 2.80$ mm. The measuring instrument is known to have a maximum absolute error of $0.005$ mm for each measurement. Calculate:

1. a) The upper bound of the absolute error made when calculating the area of the microchip.
2. b) The interval of possible values in which the actual area lies.
3. c) What is the maximum relative error made in calculating the area, expressed as a percentage rounded to three decimal places?

In a particle physics experiment, the mass of a sample is estimated to be $m = 3.45 \times 10^{-6}$ kg with a relative error of $0.2\%$. Simultaneously, the volume occupied by the sample is measured as $V = 1.2 \times 10^{-9}$ $m^3$ with an upper bound on absolute error of $0.05 \times 10^{-9}$ $m^3$.

Determine whether it is possible for the actual density ($\rho = m/V$) of the sample to exceed $3000$ $kg/m^3$, justifying your answer by calculating the corresponding error bounds.

A financial analyst uses the approximation of the golden ratio $\Phi \approx 1.618$ to predict the behaviour of a cryptocurrency on a growth graph. Given that the exact value is $\Phi = \frac{1 + \sqrt{5}}{2}$:

1. a) Calculate the absolute error made in this approximation (use all the decimal places on your calculator for the exact value).
2. b) Calculate the relative error made.
3. c) If the analyst applies this ratio to an investment of $2,500,000$ €, what would be the difference in euros between using the approximate value and the exact value? Is this error significant in a financial context?

A hiker uses a mountain GPS to find their way. The device estimates the distance to their destination shelter using the formula $d = \sqrt{x^2 + y^2}$. At a point along the route, the recorded coordinates are $x = 800$ m and $y = 600$ m. However, the GPS has an inaccuracy (upper bound on absolute error) of $2$ metres along each axis.

Calculate the upper bound of the relative error made in estimating the total distance $d$. Round the result to four significant figures.

Consider the numbers $A = 5.42 \times 10^8$ and $B = 3.1 \times 10^7$. Both numbers have been rounded to the number of significant figures shown.

1. a) Determine the upper bounds on absolute error for $A$ and $B$.
2. b) Calculate the value of $S = A + B$ and express the result in standard form.
3. c) Find the upper bound on the relative error of the sum $S$. How does this relative error compare with the relative errors of $A$ and $B$ separately?

A cylindrical fuel tank has a height of $H = 10.0$ m and a base radius of $R = 4.0$ m. The measurements were taken to a precision of one decimetre.

1. a) Calculate the nominal volume of the tank in cubic metres using $\pi \approx 3.1416$.
2. b) Determine the upper bound of the absolute error made in calculating the volume due to inaccuracy in the measurements of $R$ and $H$ (take the value of $\pi$ as exact for this part).
3. c) If the fuel expands and the actual volume is $1\%$ greater than the volume calculated from the measurements (which may deviate from the nominal value by up to the relative bound found in part b), what would be the total relative error with respect to the initial nominal value in the worst case?

The speed of a fibre optic connection has been measured, giving a value of $540.5$ Mbps. The technician states that the relative error of the measurement is less than $0.05\%$.

1. a) Between which values can the actual connection speed lie?
2. b) If we round the result to the nearest ten Mbps, what would be the new upper bound on the maximum relative error incurred with respect to the actual speed, taking into account the interval found in part a)?

Given the function $f(x) = \frac{1}{x-1}$, we want to calculate its value for $x = \sqrt{2}$.

1. a) Approximate the result using the value of $\sqrt{2}$ rounded to three decimal places.
2. b) Calculate the "exact" value (using all possible decimal places on the calculator).
3. c) Determine the absolute error and the relative error made.
4. d) Explain with reasons why, in this type of function, a small variation in the approximation of $x$ can produce a much larger relative error in the result of $f(x)$ when $x$ is close to $1$.

An e-commerce SME analyzes its commercial margin using the ratio $M = \frac{B}{I}$, where $B$ is the net profit obtained and $I$ is the total invoiced revenue. In the last quarter, a revenue of $I = 200\,000$ € is recorded with an absolute uncertainty of $\Delta I = 4\,000$ €, and a profit of $B = 10\,000$ € with an absolute uncertainty of $\Delta B = 800$ €. To estimate the interval in which the actual margin ranges, the finance team debates between two procedures:

• Method 1 (Linearization): calculate the nominal margin $M_0 = \frac{B}{I}$, add the relative error bounds to obtain $\Delta M / M_0 \approx \frac{\Delta B}{B} + \frac{\Delta I}{I}$, and define the interval as $[M_0 - \Delta M, M_0 + \Delta M]$.

• Method 2 (Interval arithmetic): calculate exactly the minimum possible margin $M_{\min} = \frac{B - \Delta B}{I + \Delta I}$ and the maximum possible margin $M_{\max} = \frac{B + \Delta B}{I - \Delta I}$.

1. a) Apply both methods detailing all calculations and express the resulting intervals in percentage form with three decimal places.
2. b) Compare the two intervals obtained: do they have the same width? Are they centered on the same value? Explain analytically what causes the discrepancy between the two methods.
3. c) If the company must present this report to a banking institution to apply for financing without risk of falsehood or overestimation, justify in a reasoned manner which of the two methods is preferable to use.

An agri-food cooperative exports extra virgin olive oil in cubic metal tanks with a nominal edge length of $a = 80$ cm. The storage volume is calculated as $V = a^3$.

The logistics department requires that the percentage relative error in the volume calculation does not exceed $0.75$ % ($E_r(V) \le 0.0075$) to avoid issues in ocean freight.

1. a) Using the first-order error propagation relation for a power ($V = a^n$), deduce the maximum allowable relative error bound in the direct measurement of the edge length $a$.
2. b) From the result of the previous section, determine the maximum tolerable absolute error bound $\Delta a$ when measuring the edge length of $80$ cm.
3. c) The manufacturer has three measuring instruments available: a common tape measure graduated in whole centimeters ($u = 1$ cm), a tape measure with half-centimeter marks ($u = 0.5$ cm), and a laser ruler calibrated to the millimeter ($u = 0.1$ cm). If the cooperative adopts rounding to the nearest multiple of the instrument's graduation as the reading criterion, determine which is the least expensive instrument (that is, the one with the lowest precision) that guarantees compliance with the logistical requirement. Prove your choice numerically.