Exercises: Approximation and errors

Round and truncate the number $\frac{17}{6}$ to thousandths. Calculate the absolute error in both approximations and state with justification which of the two is more precise.

When measuring the length of a table of $1,25$ m, a value of $1,28$ m was obtained. On the other hand, when measuring the distance between two cities of $245$ km, a value of $246$ km was obtained. Calculate the relative error in both measurements and determine which of the two is more precise.

Write the number $0,000045602$ in standard form leaving exactly three significant figures. Next, state a bound on the absolute error made when making this approximation.

The golden ratio $\Phi = \frac{1+\sqrt{5}}{2}$ is frequently approximated in graphic design by the value $1,618$. Calculate the absolute error and the relative error of this approximation, using all digits from your calculator for the exact value. Express the relative error as a percentage.

Determine the real interval containing the exact value of a measurement $x$ if we know that its approximation is $5,4$ and the absolute error bound is $E_a < 0,05$.

Perform the mathematical operation $3,2 \cdot 10^4 + 5,1 \cdot 10^3$, expressing the final result in standard form with the appropriate number of significant figures.

The irrational number $\pi$ has been approximated as $3,14$. Is it an underestimate or an overestimate? Justify your answer and find a bound for the absolute error made.

A digital thermometer shows a body temperature of $37,5^\circ\text{C}$, guaranteeing a relative error of less than $0,2\%$. What is the maximum absolute error the device could be making?

In a laboratory practical, Lucía measures the length of two objects: for the width of the classroom she obtains $L_A = 8{,}40\text{ m}$ with an absolute error bound of $2\text{ cm}$, and for the thickness of a book she obtains $L_B = 3{,}2\text{ cm}$ using a ruler whose absolute error bound is $1\text{ mm}$. When presenting her report, Lucía concludes: "The book measurement is much more precise and of higher quality because we have made an error of at most $1\text{ mm}$, whereas in the classroom the error reaches $2\text{ cm}$". Identify Lucía's conceptual error in reasoning, explain what magnitude she should have used to compare the precision of such different measurements, and calculate the percentage relative errors of both to determine which one was actually more precise.

A positive real number $x$ is approximated by rounding to the nearest tenth, obtaining the value $7{,}4$. After consulting the measurement protocol, it is known that the relative error made in said approximation is exactly $0{,}5\%$.

1. a) From the definition of relative error, set up an absolute value equation and calculate the two possible real numerical values that $x$ can take.
2. b) Check in a reasoned manner whether both values obtained strictly satisfy the rule for rounding to the nearest tenth giving $7{,}4$.

The mass of a planet is $5,97 \cdot 10^{24}$ kg and that of its natural satellite is $7,35 \cdot 10^{22}$ kg. Calculate the total mass of the planetary system in standard form and determine a bound for the absolute error of the sum, assuming that the given data are already approximations rounded to the last decimal place shown.

In a microbiology laboratory, the diameter of a cell is measured as $0,000015$ m with an absolute error of less than $10^{-7}$ m. At the same time, an astronomical observatory measures the distance to a nearby star as $4,2 \cdot 10^{16}$ m with an absolute error of less than $10^{14}$ m. Which measurement has greater actual precision? Justify your answer by calculating and comparing the relative error of both.

A rectangular room is to be tiled. The approximate measurements of the sides, taken with a tape measure, are $a = 4,2$ m and $b = 3,5$ m. Knowing that the instrument introduces a maximum error of $0,05$ m in each measurement, calculate the nominal area of the room and determine the strict interval of real values in which this area lies.

The tender budget for the construction of a viaduct is $12,4$ million euros. If, due to fluctuations in material prices, the final cost incurs a relative error of $3,5\%$, determine between which exact financial amounts (in euros) the final cost of the work will range.

Calculate the exact value of the expression $\frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}}$ by rationalising the denominator. Next, approximate the result to two decimal places and determine a bound on the relative error made with respect to the exact value.

An old geometric calculation software uses the approximation $\frac{22}{7}$ for the number $\pi$. How many exact decimal places does this approximation provide compared to the real value of $\pi$? Calculate the percentage relative error made by the software.

The speed of light in a vacuum is a physical constant whose exact value is $299792458$ m/s. Express this quantity in standard form rounded to four significant figures. What is the bound on the absolute error and the relative error of this new abbreviated representation?

When approximating a real number $x$ by the value $3,14$, an absolute error of less than $0,005$ is made. When approximating another real number $y$ by the value $2,71$, an absolute error of less than $0,005$ is also made. Determine the guaranteed error interval for the sum $x + y$ and for the difference $x - y$.

To determine the density of a geological sample using the formula $\rho = \frac{m}{V}$, its mass is measured, obtaining $m = 450\text{ g}$, rounded to the nearest ten grams, and its volume is measured by immersion in a graduated cylinder, resulting in $V = 180\text{ cm}^3$, also rounded to the nearest ten $\text{cm}^3$.

1. a) Before carrying out exhaustive calculations, make a first-order estimate of the upper bound of the percentage relative error in the density by applying the properties of relative error propagation in a quotient.
2. b) Write the intervals of possible values in which the actual mass and actual volume lie. From them, determine the exact interval $[\rho_{\min}, \rho_{\max}]$ for the density and find the maximum percentage relative error with respect to the nominal value $\rho_{\text{nom}} = \frac{450}{180} = 2{,}5\text{ g/cm}^3$.
3. c) Compare the initial estimate from part a) with the value obtained in part b) and judge whether the first-order rule is sufficiently representative in this case.

The State Meteorological Agency (AEMET) records the precipitation falling in the catchment basin of a reservoir whose surface area is estimated at $A = 650\text{ km}^2$, rounded to the nearest ten $\text{km}^2$. During a storm event, rain gauges show an average precipitation of $p = 84\text{ mm}$, rounded to the nearest millimetre ($1\text{ mm} = 1\text{ L/m}^2 = 10^{-3}\text{ m}^3/\text{m}^2$). The total volume of water that fell is modelled as $V = A \cdot p$.

1. a) Express the surface area in $\text{m}^2$, the precipitation in $\text{m}$, and calculate the nominal volume of water that fell in cubic hectometres ($1\text{ hm}^3 = 10^6\text{ m}^3$).
2. b) Determine the interval of possible values of the actual volume of water $[V_{\min}, V_{\max}]$ in $\text{hm}^3$ taking into account the rounding error bounds of the original measurements.
3. c) A spokesperson for the hydrographic confederation states to the media: "We can assure with mathematical certainty that the basin has received at least $54\text{ hm}^3$ of water". Justify analytically whether this statement is true or if there is a possibility that a smaller volume has entered.

A solid steel spherical part is manufactured for an industrial bearing. The digital calliper used indicates that the radius is $r = 12,5$ mm, with an instrumental margin of error of $\pm 0,1$ mm. Calculate the nominal volume of the sphere and analytically prove the maximum absolute error and maximum relative error propagated to the calculation of the volume.

A financial institution calculates daily interest on an account using rounding to two decimal places in each daily transaction, whereas a second institution stores all decimals internally and only rounds the total balance to two decimal places at the end of the month (consider a 30-day month). If the exact daily interest generated in the account is $1,234$ €, calculate the total accumulated absolute error differentiating both methods at the end of the month.

In a land surveying project, the two legs of a right-angled triangular plot of land are measured, giving $c_1 = 30,0$ m and $c_2 = 40,0$ m. The precision of the surveying instrument guarantees a relative error of less than $1\%$ in each measured linear length. Determine the nominal length of the hypotenuse $h$ and bound the absolute error and the maximum relative error that can be made when calculating it indirectly.

Prove algebraically that if two positive real numbers $x$ and $y$ are approximated by $x_0$ and $y_0$ with a relative error less than $\varepsilon$ in both cases, then the relative error of the product $x \cdot y$ is bounded approximately by $2\varepsilon$ (you may neglect higher-order terms above $\varepsilon$, that is, terms containing $\varepsilon^2$).

Newton's law of universal gravitation is given by the expression $F = G \frac{m_1 m_2}{d^2}$. If in an experiment we measure the masses $m_1, m_2$ and the distance $d$ with a maximum relative error of $2\%$ in each of the three quantities, and we treat the constant $G$ as an exact mathematical value, determine with justification the bound on the maximum relative error that will be made when calculating the force $F$.

We want to prepare a chemical solution in which the mass of the solute must be $m = 15,00$ g and the volume of the solvent $V = 250$ ml. The laboratory balance has a guaranteed precision of $\pm 0,01$ g and the measuring cylinder has a margin of error of $\pm 1$ ml. Calculate the nominal concentration $C = \frac{m}{V}$ expressed in g/ml and state the strict interval of real values within which this concentration lies.

We approximate the irrational value of $\sqrt{10}$ using the rational fraction $\frac{19}{6}$. Calculate the absolute error made with this approximation. Then, find a fraction with a denominator less than $20$ that provides a better approximation (that is, with a strictly smaller absolute error) and prove analytically that it is better.

A computer algorithm approximates the exponential function for values near zero using the second-degree Taylor polynomial: $e^x \approx 1 + x + \frac{x^2}{2}$. If we evaluate this algorithm at $x = 0,1$, calculate the absolute error and the percentage relative error of this approximation compared with the value given by the scientific calculator for $e^{0,1}$. How many correct significant figures does this polynomial approximation guarantee?

To calculate the internal resistance $r$ of a battery, the physical relationship $r = \frac{V_0 - V_1}{I}$ is used. The open-circuit voltage $V_0 = 12{,}4\text{ V}$, the voltage under load $V_1 = 11{,}9\text{ V}$, and the supplied current $I = 2{,}0\text{ A}$ are measured, all rounded to the last digit shown (tenths).

To estimate the uncertainty of the nominal result $r_{\text{nom}} = 0{,}25\ \Omega$, two calculation methods are proposed:

• Method 1 (First-order linear approximation): The absolute error bound of the numerator is calculated as $\Delta(V_0 - V_1) = \Delta V_0 + \Delta V_1$, and its relative error is added to the relative error of the denominator: $\varepsilon_r \approx \frac{\Delta V_0 + \Delta V_1}{V_0 - V_1} + \frac{\Delta I}{I}$.

• Method 2 (Rigorous interval arithmetic): The possible extremes of the fraction are evaluated: $r_{\max} = \frac{(V_0)_{\max} - (V_1)_{\min}}{I_{\min}}$ and $r_{\min} = \frac{(V_0)_{\min} - (V_1)_{\max}}{I_{\max}}$.

1. a) Apply both methods numerically and determine the percentage relative error bound yielded by each with respect to the nominal value.
2. b) Compare the two methods and argue which of them offers a strict and safe bound, explaining the cause of the numerical discrepancy between both.
3. c) Notice that the initial voltages have a relative error under $0{,}5\%$, but the final relative error of the resistance exceeds $20\%$. Explain what numerical phenomenon has caused this surge in error when subtracting close quantities.

In an aerospace project, a flat circular solar panel is designed. With fixed perpendicular solar incidence, the electrical power generated is modelled by $P = k \cdot R^2$, where $k = 1360\text{ W/m}^2$ is a constant known with complete accuracy and $R$ is the panel's radius in metres. The nominal power needs to be $P_{\text{nom}} = 3060\text{ W}$.

1. a) Calculate the required nominal radius $R$ for the panel.
2. b) The technical specifications require that the percentage relative error in the power does not exceed $1{,}2\%$ under any circumstances. Using the first-order propagation bound for powers ($|n|\varepsilon_R$), determine the maximum allowable tolerance (absolute error bound, in millimetres) that can be granted to the machining of radius $R$.
3. c) If the actual radius is $R(1+\delta)$ with a positive relative error $\delta > 0$, prove algebraically that the actual relative error in the power is always strictly greater than the linear approximation $2\delta$. With the tolerance bound obtained in part b) (that is, $\delta = 0{,}6\%$), calculate the error made by the first-order formula itself with respect to the actual relative error.