Exercises: Measurement of time

Look at the following image of a wall clock that uses the Roman numeral system. State the time shown on the clock, expressing the hours and minutes in the decimal system.

In the study of history and time, we use units larger than a year. Complete the following equivalences:

1. a) $3$ centuries = $\dots$ years.
2. b) $5$ decades = $\dots$ years.
3. c) $8$ lustrums = $\dots$ years.
4. d) $200$ years = $\dots$ centuries.

A short video on social media lasts exactly $145$ seconds. Express the duration of this video in compound form, that is, stating the minutes and seconds it lasts ($145$ s = $\dots$ min $\dots$ s).

This afternoon you spent two periods of time studying for the Maths exam. The first lasted $1$ h $40$ min and the second $55$ min. Calculate the total study time, expressing the result in hours and minutes.

An animated film starts at $17:15$ and ends at $18:50$. What is the exact duration of the film in hours and minutes?

Marcos is preparing for a $10\text{ km}$ fun run. To track his training, he writes down in his journal that on Monday he ran for $1\text{ h } 40\text{ min}$ and on Wednesday for $1\text{ h } 50\text{ min}$. To calculate the total accumulated time, he enters $1{,}40 + 1{,}50 = 2{,}90$ into the calculator and concludes that he has trained "$2\text{ hours and } 90\text{ minutes}$", which according to him is equivalent to "$3\text{ hours and } 30\text{ minutes}$"

1. a) Identify and explain in detail the two conceptual mistakes Marcos made when working with units of time.
2. b) Correct the calculation by properly performing the addition in the sexagesimal system and express the final result in compound units ($h$ and $min$).
3. c) How many hours as a decimal number does the total time Marcos ran actually correspond to? Round to two decimal places.

An analog wall clock shows exactly $03:20$.

1. a) Before carrying out any numerical calculations, make a reasoned estimate: will the smaller angle formed by the two hands be exactly $30^\circ$, greater than $30^\circ$, or less than $30^\circ$? Justify your prediction by thinking about the position of the hour hand when the minute hand has already moved $20\text{ minutes}$.
2. b) Calculate the exact position in degrees of each hand taking $12:00$ ($0^\circ$) as the reference point, knowing that the minute hand moves $6^\circ$ per minute and the hour hand moves $30^\circ$ per hour plus $0{,}5^\circ$ for each elapsed minute.
3. c) Determine the exact angle formed by both hands at $03:20$ and compare it with your initial estimate. Why do many people mistakenly think at first glance that at $03:20$ the hands form a $30^\circ$ angle?

An elite athlete completed a marathon in an official time of $2$ hours, $12$ minutes and $15$ seconds.

1. a) Express the total duration of the race in seconds only.
2. b) If their coach tells them to reduce their time by $450$ seconds for the next competition, what would their new target time be, expressed in compound form ($h$, $min$, $s$)?

The high-speed train (AVE) leaves Madrid at $08:45:00$ and is scheduled to arrive in Seville at $11:17:25$. Due to a minor technical delay, the train arrived at $11:23:10$.

1. a) What was the scheduled duration of the trip?
2. b) How much delay did the train experience compared to the scheduled time?

A youth association organises a park clean-up day that lasts a total of $5$ hours, $42$ minutes and $36$ seconds. If $4$ work teams have been formed and they want to split the total time equally so that each team covers a different time slot without overlapping:

How long should each team's shift last? (Carry out the division in the sexagesimal system).

The production team of an educational television channel has a documentary that lasts exactly $1\text{ h } 14\text{ min } 24\text{ s}$ and needs to divide it into $6$ blocks of identical duration to insert educational breaks.

1. a) Apply Method A (direct division in sexagesimal compound units): divide the hours by 6, convert the remainder into minutes, add them to the existing minutes, and repeat the process with the seconds.
2. b) Apply Method B (reduction to the smallest unit): convert the entire original duration into seconds, carry out the division by 6 in decimal base, and convert the resulting quotient back into hours, minutes, and seconds.
3. c) Compare both procedures: with which of the two methods do you consider there is less risk of making carrying errors? If the duration of the documentary had to be shared among $7$ teams instead of $6$, which of the two methods would be more advantageous and why?

A high-speed AVE train departs from Madrid-Puerta de Atocha station at $08:35$ heading to Sevilla-Santa Justa. According to the official scheduled itinerary, the duration of the direct journey without incidents is $2\text{ h } 42\text{ min } 30\text{ s}$. During the trip, two events occur:

• While passing through La Mancha, it undergoes a temporary speed restriction due to construction work that adds a delay of $14\text{ min } 45\text{ s}$.

• On the final stretch between Córdoba and Sevilla, the train makes up $3\text{ min } 15\text{ s}$ en route.

1. a) Model the situation using operations in the sexagesimal system and calculate the exact time (hours, minutes, and seconds) at which the train finally arrives in Sevilla.
2. b) The length of the railway line traveled is $471\text{ km}$. Express the actual total duration of the trip in seconds and, from that data, calculate the average speed of the train throughout the entire journey in kilometers per hour ($km/h$), rounding the result to the nearest tenth.

An ultra-trail runner has completed a $105 \text{ km}$ race in a time of $14 \text{ h } 18 \text{ min } 36 \text{ s}$.

1. a) Calculate their average speed in metres per second ($m/s$), rounding to hundredths.
2. b) If the course record is $12 \text{ h } 45 \text{ min } 50 \text{ s}$, by how many seconds per kilometre would they have needed to reduce their time to match the record?

An analogue clock shows exactly $10:15:30$ (10 hours, 15 minutes and 30 seconds). Calculate the exact angle formed by the hour hand and the minute hand at that precise moment. Note: Take into account that the hour hand moves continuously as the minutes and seconds pass.

The planet Mars has a rotation period (a "sol") of approximately $24 \text{ h } 37 \text{ min } 22 \text{ s}$. An astronaut arrives on Mars and starts an Earth stopwatch and a stopwatch set to Martian time right at sunrise. What Earth time difference will there be between both clocks when exactly one Martian week (7 sols) has elapsed? Express the result in compound form (hours, minutes and seconds).

An internet server processes user requests. It is known to take an average of $1,25$ hundredths of a second to handle each request. If during a cyberattack the server receives a total of $2^{18}$ requests:

1. a) Calculate the total number of requests in the decimal system.
2. b) Calculate the total time it will take for the server to process them all. Express the final result in hours, minutes and seconds.

On an analog clock, both hands coincide exactly at $12:00$ (forming an angle of $0^\circ$). However, throughout the morning they overlap again several times. We want to find out when this happens between $04:00$ and $05:00$.

1. a) If we call $m$ the number of minutes elapsed since $04:00$ sharp, write the algebraic expression for the angular position (in degrees from $12:00$) of the hour hand and the minute hand as a function of $m$. Remember that at 4 o'clock sharp the hour hand starts at $120^\circ$ and advances $0{,}5^\circ$ per minute, while the minute hand advances $6^\circ$ per minute.
2. b) Set up the equation stating that both hands coincide at the same angular position and calculate the exact value of $m$. Express the solution as an irreducible fraction of a minute and as a mixed number ($m = A + \frac{B}{C}$).
3. c) From that value, determine the exact time of the coincidence expressed in full hours, minutes, and seconds (rounding the seconds to the nearest whole number). Verify numerically that at this calculated time the angular positions of both hands differ by less than $0{,}1^\circ$.

A photovoltaic solar plant in the province of Cáceres (Extremadura) has panels equipped with a tracking axis that rotate from east to west by a total angle of $180^\circ$ to stay oriented perpendicularly to the Sun. According to local astronomical records, at the summer solstice the Sun rises at $06:52$ and sets at $21:48$, while at the winter solstice it rises at $08:38$ and sets at $18:04$.

1. a) Model and calculate the length of the day (daylight time) at each solstice in compound units ($h$ and $min$). How many hours and minutes of daylight does the plant lose at the winter solstice compared to the summer one?
2. b) In order for the panel to continuously and uniformly complete the $180^\circ$ path exactly during the daylight hours, calculate the constant angular velocity (in degrees per hour, rounding to two decimal places) that must be programmed into the motor at the summer solstice and at the winter solstice.
3. c) The maintenance manager proposes a simplification: "Since the Earth rotates $360^\circ$ in 24 hours, the Sun always moves across the sky at $15^\circ$ per hour. Let's keep the motor programmed all year round at a fixed speed of $15^\circ$ per hour". Mathematically prove how many degrees of mismatch the panel would accumulate at sunset during the winter solstice if his proposal were followed, and explain why his theoretical approach is not valid for this tracking mechanism.

Unit challenge. How does the duration of sunlight change throughout the year? Investigate the solar rhythm of your municipality through time measurement and the sexagesimal system.

1. Collection: Check the official IGN (ign.es) or AEMET website for the exact sunrise and sunset times in your town on two dates: the past 21 June and 21 December.

2. Sexagesimal calculation: Calculate by hand the duration of daylight for both days by subtracting the sunrise time from the sunset time in sexagesimal format ($h$, $min$, $s$), applying base-$60$ conversions when borrowing.

3. Digital processing: In a spreadsheet, record the data and convert both durations to decimal hours (for example, $9\text{ h } 30\text{ min} = 9,5\text{ h}$). Find the total difference in daylight between both dates and calculate the average daily variation by dividing that difference by the $183$ elapsed days.

4. Final report: Write a reasoned half-page conclusion evaluating the pros and cons of using the sexagesimal system versus the decimal system to measure time in everyday life.

The following will be assessed:

• Rigour in operations and conversions in the sexagesimal system.

• Correctness in converting sexagesimal expressions to decimals (with a comma).

• Proper use of formulas in the spreadsheet.

• Critical reasoning in the conclusions of the report.