Exercises: Approximation and errors

Round and truncate the number $\frac{17}{6}$ to three decimal places. 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 $1,25$ ft, a value of $1,28$ ft was obtained. On the other hand, when measuring the distance between two cities of $245$ miles, a value of $246$ miles was obtained. Calculate the relative error in both measurements and determine which of the two is more precise.

Write the number $0,000045602$ in scientific notation leaving exactly three significant figures. Then, state a bound for 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 the figures on your calculator for the exact value. Express the relative error as a percentage.

Determine the real interval in which the exact value of a measurement $x$ is found 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 scientific notation with the appropriate number of significant figures.

The irrational number $\pi$ has been approximated by taking the value $3,14$. 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 that the device could be making?

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 scientific notation and determine a bound for the absolute error of the sum, assuming that the data provided are already approximations rounded to the last decimal figure shown.

In a microbiology laboratory, the diameter of a cell is measured to be $0,000015$ m with an absolute error of less than $10^{-7}$ m. In parallel, 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 $a = 4,2$ ft and $b = 3,5$ ft. Knowing that the instrument introduces a maximum error of $0,05$ ft in each measurement, calculate the nominal area of the room and determine the strict interval of real values in which this area is found.

The tender budget for the construction of a viaduct is $12,4$ million dollars. If, due to fluctuations in the price of materials, the final cost suffers a relative error of $3,5\%$, determine between which exact economic amounts (in dollars) the final cost of the work will fluctuate.

Calculate the exact value of the expression $\frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}}$ by rationalizing the denominator. Then, approximate the result to two decimal places and determine a bound for 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 scientific notation rounding to four significant figures. What is the bound of 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 caliper used indicates that the radius is $r = 12,5$ in, with an instrumental margin of error of $\pm 0,1$ in. Calculate the nominal volume of the sphere and prove analytically what the maximum absolute error and the maximum relative error propagated to the volume calculation are.

A financial institution calculates the daily interest on an account using rounding to two decimal places in each daily operation, while a second institution internally stores all the decimal places 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 that differentiates both methods at the end of the month.

In a surveying project, the two shorter sides of a right triangular plot of land are measured, obtaining $c_1 = 30,0$ ft and $c_2 = 40,0$ ft. 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 find the bounds for 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 of less than $\varepsilon$ in both cases, then the relative error of the product $x \cdot y$ is bounded approximately by $2\varepsilon$ (you can neglect terms of order higher than $\varepsilon$, 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 $d$ making a maximum relative error of $2\%$ in each of the three quantities, and we consider the constant $G$ as an exact mathematical value, determine and justify the bound of the maximum relative error that will be made when calculating the force $F$.

A chemical solution is to be prepared in which the mass of the solute must be $m = 15,00$ oz and the volume of the solvent $V = 250$ fl oz. The laboratory balance has a guaranteed precision of $\pm 0,01$ oz and the graduated cylinder has a margin of error of $\pm 1$ fl oz. Calculate the nominal concentration $C = \frac{m}{V}$ expressed in oz/fl oz 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. Next, 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 so.

A computer algorithm approximates the exponential function for values close to 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 to the value provided by a scientific calculator for $e^{0,1}$. How many correct significant figures does this polynomial approximation guarantee?