Exercises: Approximations

Round the number $\pi \approx 3,14159265...$ to the nearest tenth, hundredth and thousandth.

Truncate the irrational number $\sqrt{7} \approx 2,645751...$ to two decimal places and calculate the absolute error made with this approximation.

When measuring the length of a table of exactly $2,5 \text{ m}$, a tape measure gave a value of $2,46 \text{ m}$. Calculate the absolute error and the relative error of this measurement.

State the number of significant figures in the following experimental measurements: $0,0045 \text{ kg}$, $30,020 \text{ s}$, $1500 \text{ m}$ (assuming it is exact to the nearest unit) and $2,0 \cdot 10^5 \text{ A}$.

Express the number $45678,9$ to three significant figures, using standard form.

A bathroom scale reads $54,2 \text{ kg}$ when weighing a person. If we know from a precision scale that the exact value is $54,25 \text{ kg}$, determine the relative error expressed as a percentage.

When an unknown number is rounded to two decimal places, $4,38$ is obtained. State the interval containing the exact value of the number and determine the bound on the absolute error.

Approximate the fraction $\frac{17}{7}$ to three decimal places by overestimate and underestimate.

Marcos is analysing the results of a tap water quality analysis from his municipality and reads that the nitrate concentration is $0{,}040\text{ mg/l}$. While preparing a presentation for class, Marcos states: "This value has four significant figures because there are four digits in total, and since it was measured to the thousandths, the bound on the absolute error is $0{,}005\text{ mg/l}$". Find the two errors made by Marcos, explain in detail why his reasoning is incorrect in each case, and write the corrected values for both the number of significant figures and the actual bound on the absolute error of the measurement.

According to data from the Continuous Municipal Register of the Instituto Nacional de Estadística (INE), the city of Teruel has an official population of $36\,267$ inhabitants. Two different approximations appear in the media: a general newspaper runs the headline "Teruel has around $36\,000$ inhabitants", while a municipal technical report prefers to use the figure of $36\,300$ inhabitants.

1. a) Identify to which order of magnitude (tens, hundreds, or thousands) each of the two sources has rounded.
2. b) Write the technical report's figure ($36\,300$) in scientific notation so that it clearly expresses that it has three significant figures.
3. c) Calculate the absolute error made by each publication with respect to the official INE data and determine which of the two approximations guarantees a bound on the absolute error of less than $50$ inhabitants.

In a supermarket, the exact price by weight of a piece of cheese is $14,53 €$ and at the till $14,50 €$ is charged. On the other hand, a loaf of bread costs exactly $0,83 €$ and $0,80 €$ is charged. Calculate the relative error in both purchases and state in which of them the error affects the consumer more in proportion to what they spend.

The diameter of a human red blood cell is approximately $7,8 \cdot 10^{-6} \text{ m}$. The software of an electron microscope approximates this measurement and displays $0,000008 \text{ m}$ on the screen. Calculate the absolute error and the percentage relative error made by the software. How many significant figures does the value displayed on the screen have?

To construct a perfect square frame, the interior area needs to be exactly $12 \text{ m}^2$. The carpenter calculates the required side length and rounds it to two decimal places to cut the wooden strip. What will be the absolute error in the total perimeter of the constructed frame compared to the ideal perimeter?

The road distance between two cities is published as $340 \text{ km}$, stated to have two significant figures. Determine the interval of values containing the actual distance and calculate the bound on the absolute error and the maximum possible relative error bound.

A text file has an exact size of $1456783 \text{ bytes}$. The operating system displays its size in the folder as $1,4 \text{ MB}$ (assuming for this exercise that $1 \text{ MB} = 10^6 \text{ bytes}$). Calculate the absolute error and the relative error made by the operating system's approximation.

In a school athletics competition, the exact time of a runner crossing the finish line is $10,458 \text{ s}$. Judge A's stopwatch works by truncating to two decimal places, whereas Judge B's stopwatch rounds to two decimal places. Calculate the absolute error of each stopwatch and explain which of the two judges gives a time closer to the real time.

The universal gravitational constant has an approximate value of $G \approx 6,674 \cdot 10^{-11} \text{ N}\cdot\text{m}^2/\text{kg}^2$. Round this value so that it has only three significant figures and find the bound on the relative error of this new approximation.

A pharmacy digital thermometer reads $37,5^\circ\text{C}$. The manufacturer's manual states that the sensor's maximum relative error is $0,4\%$. Calculate the interval of temperatures in which the patient's actual temperature definitely lies.

In a $25\text{ km}$ time-trial stage of La Vuelta Ciclista a España, the organisation's official stopwatch records a time of $34\text{ min and } 18{,}4\text{ s}$ for the winner, a value rounded to the nearest tenth of a second.

1. a) Express the official time solely in seconds. What is the bound on the absolute error associated with that time measurement? Write the interval $[t_{\min}, t_{\max}]$ in seconds in which the cyclist's actual time definitely lies.
2. b) During the television broadcast, a commentator rounds the total time to whole minutes, stating that the rider took "$34\text{ minutes}$". Calculate the absolute error and the percentage relative error made by the commentator when using that approximation with respect to the official figure in seconds. Assess whether his approximation is reasonable for the television audience.

You are going shopping at a supermarket to prepare a family meal on a tight budget. On your list, there are three items:

• $1{,}8\text{ kg}$ of chicken at $6{,}95\text{ €/kg}$.

• $0{,}85\text{ kg}$ of cheese at $11{,}20\text{ €/kg}$.

• $2{,}1\text{ kg}$ of apples at $1{,}89\text{ €/kg}$.

In your wallet, you only have a $30\text{ €}$ banknote.

1. a) Before calculating exactly, make a quick estimate of the total amount by rounding each quantity in kilograms to the nearest whole unit and each unit price to the nearest whole euro. Based on that mental estimate, determine whether it seems you will have enough money to pay.
2. b) Now perform the exact calculation of the total cost by adding the prices of each item rounded to the nearest euro cent (as the cash register scanner does).
3. c) Calculate the absolute error made between your initial estimate and the final checkout price. In light of the result, reflect: was your estimate an overestimate or an underestimate? Why, when shopping on a fixed budget, is it usually advisable to deliberately overestimate?

In the construction of a bridge, an engineer measures the cross-sectional dimensions of a rectangular beam, obtaining $L = 15,4 \text{ m}$ and $A = 0,85 \text{ m}$. Both measurements have been rounded to the last decimal place shown. Determine the interval containing the actual cross-sectional area of the beam. What is the absolute error bound assumed when calculating the area by multiplying the approximate values directly?

A circular town square is to be fenced off. The surveyor measures the radius and obtains $R = 45,2 \text{ m}$ (a value rounded to one decimal place). Calculate the nominal perimeter of the square using $\pi \approx 3,1416$ and find the maximum absolute error (error bound) that can be made in calculating this perimeter due to the uncertainty in the radius measurement.

The density of a material is calculated using the formula $d = \frac{m}{V}$. If the mass is approximated to $m = 45 \text{ kg}$ with an absolute error bound of $0,5 \text{ kg}$, and the volume is approximated to $V = 2,5 \text{ m}^3$ with an absolute error bound of $0,1 \text{ m}^3$, calculate the maximum and minimum possible values for the exact density of the material.

In a laboratory, measuring instrument A has a constant absolute error of $0,05 \text{ cm}$ regardless of what it measures. Instrument B measures with a constant relative error of $0,2\%$. Determine algebraically the interval of exact lengths for which instrument A is more precise (makes a smaller absolute error) than instrument B.

Show with clear reasoning that if a real number $x \in [10, 100]$ is rounded to the nearest integer to obtain an approximation $x'$, the maximum possible relative error occurs when $x = 10,5$. What is the value of this maximum relative error expressed as a percentage?

The velocity of a projectile is measured as $v = 345 \text{ m/s}$ (to three significant figures). Its mass is estimated as $m = 12,4 \text{ kg}$ (also to three significant figures). Given that kinetic energy is calculated as $E_c = \frac{1}{2} m v^2$, calculate the approximate value of the energy, determine the appropriate number of significant figures to express the final result and estimate the bound on the relative error of the kinetic energy.

A computer stores real numbers using a floating-point standard that truncates the significand to store the equivalent of $7$ significant decimal figures. If the computer must perform and store the result of the calculation $10^8 + 1$, what absolute error and relative error does the system make when saving the final number?

An investor buys a block of shares for an exact value of $15000 €$. The broker's computer system rounds the daily percentage gain to an integer before applying it. If the actual daily gain was exactly $1,49\%$ for $5$ consecutive days, calculate the difference in euros between the final balance calculated with the exact percentage and the final balance calculated with the rounded percentage. What is the final relative error on the total balance?

An automatic weather station of AEMET located in a mountain pass records wind speed using a digital sensor. The information screen shows that the maximum recorded gust was $120\text{ km/h}$, specifying that this is an approximation obtained by rounding the measurement to two significant figures.

1. a) Determine the numerical interval $[v_{\min}, v_{\max})$ in which the actual wind speed must lie so that, when rounded to two significant figures, it gives exactly $120\text{ km/h}$. What is the bound on the absolute error assumed by any user who consults this data?
2. b) When checking the anemometer's internal log, it is observed that the device records data with one decimal place. It is known that the absolute error made with respect to $120\text{ km/h}$ was strictly less than $2\text{ km/h}$ and that, when rounding that original measurement to the nearest whole unit, the result ended in the digit $9$. Reconstruct with reasoning the interval of values in which the sensor's original reading must have been, and propose three specific numerical values with one decimal place that satisfy all the conditions.

An irrigation community in the Huerta de Murcia needs to issue the quarterly water bill for an agricultural holding. The main volumetric meter has recorded an exact consumption of $12\,640\text{ m}^3$, and the price set by the irrigation board is $0{,}276\text{ €/m}^3$. Three methods are being discussed to issue the bill:

• Method 1: Previously round both the volume and the price to 2 significant figures, and then multiply both rounded values.

• Method 2: Truncate the volume to the nearest thousand ($12\,000\text{ m}^3$) and multiply by the exact price, leaving the difference in cubic metres for an annual adjustment.

• Method 3: Multiply the exact volume by the exact unit price and round only the resulting final amount to the nearest euro cent.

1. a) Calculate the amount to pay that would result from applying each of the three methods.
2. b) Taking the result of Method 3 as the exact billing reference, calculate the absolute error and the percentage relative error made with Method 1 and with Method 2.
3. c) Compare the methods and argue: why, from a mathematical and error-propagation standpoint, is Method 1 not advisable? What economic consequences would the irrigators or the community face if the use of Method 1 were adopted instead of Method 3?