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.

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.

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.

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.

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.

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.

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 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?

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 $E_a < 0,05$.

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 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.

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.

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 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.

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 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 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.

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.

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.

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.

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?

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.

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.