Exercises: The sexagesimal number system

The sexagesimal number system is positional and has a base of $60$. State whether the following statements are true or false:

• a) In the sexagesimal system, $60$ units of one order form one unit of the next higher order.

• b) The sexagesimal system is currently used to measure temperature.

• c) One hour has $3600$ seconds.

An architect is designing a plan and needs to work with very precise angles. If they have an angle of $45^\circ$ and $20'$, express it entirely in minutes (single-unit form).

Lucía is practicing with her telescope and has turned the lens through an angle of $120^\circ$. If she then decides to reverse the turn by $35^\circ$ $15'$, what angle does her telescope show now?

In an athletics race, the first-place runner reached the finish line with a time of $1$ min $42$ s. The runner in second place arrived $15$ seconds later. What time did the second athlete record? (Express the result in minutes and seconds).

In an esports championship, the first game of a final lasted $45$ min and $38$ s. After a short break, the second game lasted $52$ min and $45$ s. Calculate the total duration of both games combined and express the result in an appropriate time format.

A professional diving team has an air supply that allows them to stay underwater for $3$ hours and $15$ minutes. If they must divide this time equally to explore $5$ different sectors of a reef, how much time should they spend observing each sector?

In a technical drawing class, we are asked to draw the complementary angle of a given angle. If the original angle measures $34^\circ 52' 18''$, carry out the necessary calculations to find the exact measure of the angle needed to complete the right angle ($90^\circ$).

The ancient Babylonians used a positional base-$60$ number system. To represent their astronomical calculations, they used symbols for $10$ ($\langle$) and for $1$ ($\curlyvee$). If an astronomer recorded the observation time of a star as the sexagesimal number $(12, 45, 18)$, where the first value is hours, the second minutes and the third seconds:

1. a) Represent this number using Babylonian symbols (grouping the wedge marks for each position).

1. b) Calculate how many total seconds the observation lasted, expressing the result in standard form.

A surveillance drone carries out a security sweep by rotating on its axis. It starts at a position of $0^\circ$ and makes the following consecutive movements:

1. Turns to the right through an angle of $125^\circ \ 45' \ 52''$.

2. Turns to the left by $87^\circ \ 50' \ 15''$.

3. Makes a third turn to the right that is exactly three times the angle resulting from subtracting the previous two movements.

Calculate the final position of the drone relative to its starting point in degrees, minutes and seconds.

A video editor is working on a short film for social media. The project has a strict time limit of exactly $10$ minutes. They have four clips with the following durations:

• Clip 1: $2$ min $14$ s $45$ hundredths.

• Clip 2: $155$ s $80$ hundredths.

• Clip 3: $3$ min $12$ s.

• Clip 4: $1$ min $58$ s $95$ hundredths.

Calculate the exact time left over or needed to reach the limit of $10$ minutes. (Remember that $1$ second has $100$ hundredths).

In the design of a part for a high-precision engine, an angle of $145^\circ \ 25'$ must be divided into 6 equal sectors in order to place some bearings. Due to machine limitations, the measure of each sector must be a whole number of minutes (no seconds) and as large as possible. When carrying out the division, the leftover angle is shared equally between two lateral safety margins.

The drawing is not to scale: the margins are so small that they cannot be seen.

1. a) Calculate the measure of each of the 6 sectors.

1. b) Calculate the leftover angle and the measure of the angle of each safety margin.

A telecommunications satellite orbits the Earth. On their control panel, a technician observes that the satellite has completed $4.72$ orbits in the last $24$ hours.

1. a) Express the decimal part of the orbits ($0.72$) in compound unit form using degrees, minutes and seconds (knowing that a complete orbit is $360^\circ$).

1. b) If the satellite takes the same time for each orbit, what is the exact time it takes to complete a single turn? Express the result in hours, minutes and seconds.

A lighthouse emits three different light signals to guide ships:

• The red light flashes every $18$ seconds.

• The green light flashes every $45$ seconds.

• The white light flashes every $1$ minute and $15$ seconds.

If at $22:00:00$ at night all three lights flash together for the first time:

1. a) How much time will pass until they flash together again? Express the result in minutes and seconds.
2. b) At what exact time will they flash together for the second time?
3. c) How many times will all three lights flash together from $22:00:00$ to $00:00:00$, both times included?

In an irregular hexagon, the measurements of five of its interior angles are known:

• $A = 110^\circ \ 15' \ 20''$

• $B = 125^\circ \ 40'$

• $C = 98^\circ \ 25''$

• $D = 130^\circ \ 12' \ 15''$

• $E = 115^\circ \ 50' \ 30''$

Given that the sum of the interior angles of a $n$-sided polygon is $S = (n - 2) \times 180^\circ$, calculate the measure of the sixth angle $F$ in compound form.