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 $3.2 \times 10^4 + 5.1 \times 10^3$, 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?

The mass of a planet is $5.97 \times 10^{24}$ kg and that of its natural satellite is $7.35 \times 10^{22}$ 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 $4.2 \times 10^{16}$ 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 $248$ million pesos. If, due to fluctuations in material prices, the final cost incurs a relative error of $3.5\%$, determine between which exact financial amounts (in pesos) the final cost of the project 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 $x - y$.

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 $1.234$ pesos, 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 $x \times y$ 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?