Approximation and errors

A Formula 1 driver has recorded a lap time of $1,24572$ minutes. To simplify the graphics in the television broadcast, approximations of this time are required.

Calculate:

1. a) The time truncated to the thousandths.

1. b) The time rounded to the thousandths.

1. c) The time rounded to the hundredths.

1. d) Briefly explain why the result of truncating and the result of rounding to the thousandths are different in this particular case.

In an electronics shop, the actual price of a smartphone is $456,28$ €. However, on the shop window advertising sign, the owner decides to display the price rounded to the nearest unit to make it more visually appealing.

Calculate:

• The approximate value displayed on the sign.

• The absolute error made with this approximation.

• The relative error expressed as a percentage (to two decimal places).

When measuring the thickness of a textbook with a ruler whose smallest division is millimetres, we obtain a value of $2,4$ cm.

State:

• The absolute error bound of this measurement.

• Between which values the actual thickness of the book lies, expressing the result as an interval.

We have two different measurements made by different operators:

1. A carpenter measures a wooden slat of $2,5$ metres and makes an absolute error of $1$ cm.

2. A surveyor measures a plot of land of $450$ metres and makes an absolute error of $1$ metre.

Calculate the relative error of each measurement and give reasons to determine which of the two approximations is of better quality (more precise).

Two sound technicians are calibrating equipment for a music festival. The first technician measures the length of an antenna cable and obtains $15,25$ m, when its actual value is $15,30$ m. The second technician measures the latency of a digital signal and obtains $0,018$ s, the actual value being $0,020$ s.

1. a) Calculate the absolute error made by each technician.
2. b) Determine which of the two measurements is more precise by comparing their relative errors.
3. c) If the first technician rounds their measurement to one decimal place, what would be the new relative error made?

The golden ratio, represented by the Greek letter $\phi$, has an exact value of $\frac{1 + \sqrt{5}}{2}$. A student uses a calculator and finds its value to be approximately $1,6180339...$

If the student decides to take $1,62$ as an approximation of the golden ratio:

• Is it a rounding or a truncation to hundredths?

• Calculate the absolute error made (take the calculator value as the true value).

• Is this absolute error less than one thousandth?

An electronic component manufacturing company produces resistors that must have a nominal value of $470$ ohms ($\Omega$). Quality control permits a margin of error (tolerance) such that the actual value $R$ must satisfy the condition $|R - 470| \leq 15$.

1. a) Express the permitted resistance values as an interval on the real number line.
2. b) If a measured resistor gives a value of $482,5 \Omega$, is it within the tolerance margin?
3. c) Calculate the upper bound on the maximum relative error accepted by the company for this component.

The radius of a circular skating rink has been measured with a tape measure, obtaining a value of $12,4$ metres. It is known that the absolute error made in this measurement is, at most, $0,05$ metres.

1. a) Write the interval in which the actual value of the radius lies.
2. b) Calculate the maximum and minimum possible values for the area of the rink (use $\pi \approx 3,14159$).
3. c) If we take as an approximation of the area the value calculated with the radius of $12,4$ m, what is the upper bound on the absolute error for that area?

In a biology laboratory, the number of bacteria on a culture plate is estimated to be $4,25 \times 10^7$, with a relative error of $2\%$.

1. a) Calculate the absolute error made in the estimation.
2. b) Between which two values does the actual number of bacteria lie?
3. c) If after one hour the number of bacteria triples, express the new total in standard form and calculate its order of magnitude.

A high-precision microchip has a rectangular base whose dimensions have been measured with a laser calliper. The length obtained is $l = 4,25$ mm and the width is $w = 2,80$ mm. The measuring instrument is known to have a maximum absolute error of $0,005$ mm for each measurement. Calculate:

1. a) The upper bound of the absolute error made when calculating the area of the microchip.
2. b) The interval of possible values in which the actual area lies.
3. c) What is the maximum relative error made in calculating the area, expressed as a percentage rounded to three decimal places?

In a particle physics experiment, the mass of a sample is estimated to be $m = 3,45 \cdot 10^{-6}$ kg with a relative error of $0,2\%$. Simultaneously, the volume occupied by the sample is measured as $V = 1,2 \cdot 10^{-9}$ $m^3$ with an upper bound on absolute error of $0,05 \cdot 10^{-9}$ $m^3$.

Determine whether it is possible for the actual density ($\rho = m/V$) of the sample to exceed $3000$ $kg/m^3$, justifying your answer by calculating the corresponding error bounds.

Consider the real numbers $A = \sqrt{7}$ and $B = \frac{22}{8}$.

1. a) Represent the number $A$ exactly on the number line using Pythagoras' theorem.
2. b) Approximate both numbers by rounding to three decimal places.
3. c) What relative error is made when approximating the sum $A + B$ by the number $5,4$?

A financial analyst uses the approximation of the golden ratio $\Phi \approx 1,618$ to predict the behaviour of a cryptocurrency on a growth graph. Given that the exact value is $\Phi = \frac{1 + \sqrt{5}}{2}$:

1. a) Calculate the absolute error made in this approximation (use all the decimal places on your calculator for the exact value).
2. b) Calculate the relative error made.
3. c) If the analyst applies this ratio to an investment of $2.500.000$ €, what would be the difference in euros between using the approximate value and the exact value? Is this error significant in a financial context?

A hiker uses a mountain GPS to find their way. The device estimates the distance to their destination shelter using the formula $d = \sqrt{x^2 + y^2}$. At a point along the route, the recorded coordinates are $x = 800$ m and $y = 600$ m. However, the GPS has an inaccuracy (upper bound on absolute error) of $2$ metres along each axis.

Calculate the upper bound of the relative error made in estimating the total distance $d$. Round the result to four significant figures.