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.

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.

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.

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.

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.

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 $E_a < 0,05$.

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?

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 $x - y$.

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 $F$.

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.

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.

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.