Metric Units, Time, and Angle Measure

Indicate which quantity (length, mass, capacity, area, or volume) is being measured in each of the following cases and suggest the most appropriate International System (SI) unit to express it:

1. a) The weight of a backpack full of books.
2. b) The amount of water that fits in an Olympic swimming pool.
3. c) The distance traveled by a high-speed train between Boston and New York.
4. d) The size of a poster you want to stick on your bedroom wall.
5. e) The space occupied by a shoebox inside a closet.

Perform the following length and mass unit conversions, moving the decimal point as appropriate:

1. a) Convert $4,52$ $km$ to meters ($m$).
2. b) Convert $830$ $mg$ to decigrams ($dg$).
3. c) Express $12,5$ $hm$ in decameters ($dam$).
4. d) Transform $0,075$ $kg$ into grams ($g$).

In the supermarket, a $2,5$ $kg$ bag of oranges costs $\$3,75$. If we want to buy a different amount, it is useful to know the price per unit. Calculate:

1. a) How many grams ($g$) of oranges are there in the bag?
2. b) What is the price of a single kilogram ($kg$) of these oranges?

A hiking club is signposting a route on the Appalachian Trail. The total journey is $12,4\text{ km}$. If they have already installed posts every $80\text{ dam}$ and they still have to cover the last $3.600\text{ m}$, how many kilometers have they already signposted? Express the final result in kilometers, hectometers and meters (complex form).

In a chemistry laboratory, three substances have been mixed for an experiment. The first substance weighs $0,25\text{ kg}$, the second $480\text{ cg}$ and the third $15,4\text{ g}$.

1. a) Calculate the total mass of the mixture in grams.
2. b) If we want to divide the mixture into $12$ equal test tubes, how many milligrams will there be in each tube?

Remember the fundamental relationship between capacity and volume units: $1$ $L = 1$ $dm^3$. Use this equivalence to solve:

1. a) A soft drink has a volume of $330$ $cm^3$. How many milliliters ($mL$) does it contain?
2. b) A water tank has a capacity of $2$ $kL$. How many cubic meters ($m^3$) does that volume occupy?
3. c) If a $1,5$ $L$ water bottle is full of distilled water, what is its exact mass in kilograms ($kg$)?

A cuboid-shaped aquarium has dimensions of $80\text{ cm}$ long, $0,4\text{ m}$ wide and $5\text{ dm}$ high. If the aquarium is filled to $\frac{4}{5}$ of its total capacity:

1. a) How many liters of water does it contain?
2. b) Knowing that $1\text{ liter}$ of water weighs $1\text{ kg}$, what is the mass of the water in the aquarium expressed in decagrams ($dag$)?

A farmer from California has produced $3,5\text{ toneladas}$ of extra virgin olive oil. For its distribution, he wants to bottle the production in $5\text{ liters}$ jugs.

1. a) Knowing that the density of the oil is $0,92\text{ kg/L}$ (that is, $1\text{ liter}$ of oil weighs $920\text{ g}$), how many liters of oil does he have in total?
2. b) How many $5\text{ L}$ jugs will he be able to fill completely? How many centiliters will be left over?

In a chemistry laboratory, they have a tank with $0,075\text{ m}^3$ of a special oil whose density is $0,92\text{ kg/dm}^3$. They want to bottle it in small flasks that have a capacity of $125\text{ ml}$.

1. a) How many full flasks will they be able to fill?
2. b) What will be the total mass, expressed in decagrams ($dag$), of all the oil contained in the tank?

Calculate the area of the following figure by counting the squares that form it. Each square on the grid represents $1$ $cm^2$. Express the final result also in square millimeters ($mm^2$).

A technology company wants to install solar panels on the roof of its warehouse. The available area is $0,15\text{ hm}^2$ and $4,2\text{ dam}^2$. Each solar panel occupies an area of $25.000\text{ cm}^2$. What is the maximum number of panels they can place if they must leave an area of $12\text{ m}^2$ free for maintenance?

We want to fence and cover with artificial grass a rectangular enclosure whose area is $0,18\text{ hectares}$. We know that one of the sides measures $36\text{ dam}$.

1. a) Calculate the length of the other side in meters.
2. b) If the price of the fence is $12,40\text{ \$}$ per linear meter and the grass is $22,50\text{ \$}$ per square decameter, what is the total cost of the materials?

A city council wants to renew the paving of a circular square that has a radius of $15\text{ m}$. The cost of the granite is $45,50\text{ \$}$ per square meter, but $21\%$ sales tax must be added to the final price. In addition, it is estimated that $5\%$ of the material is wasted in cuts. Calculate the total budget needed for the work, rounding to the nearest cent. (Take $\pi \approx 3,1416$).

The sexagesimal system is based on the number $60$. Perform the following time conversions:

1. a) How many minutes are $3$ hours and $15$ minutes?
2. b) A TikTok video lasts $150$ seconds. Express it in minutes and seconds (complex form).
3. c) If a television series lasts $0,5$ hours, how many minutes is that?

Express the following durations in seconds and then order them from greatest to least:

• $t_1 = 2,5\text{ hours}$

• $t_2 = 145\text{ minutes and } 30\text{ seconds}$

• $t_3 = 9.200\text{ seconds}$

• $t_4 = 2\text{ h } 30\text{ min } 15\text{ s}$

Emma started studying at $17:45$ and finished at $19:10$.

1. a) How much time has she been studying in total? (Express the result in hours and minutes).
2. b) If she took a $15$-minute break in the middle of the afternoon, how much real study time has she had?

An eSports tournament consists of three stages. The first stage lasted $2\text{ h } 45\text{ min } 15\text{ s}$, the second stage lasted $1\text{ h } 58\text{ min } 50\text{ s}$ and the grand final lasted $55\text{ min } 20\text{ s}$.

1. a) What was the total duration of the tournament in complex form?
2. b) If the tournament was recorded in its entirety and the video file takes up $12\text{ MB}$ for each minute of recording, how many $GB$ does the total video take up approximately? (Fact: $1\text{ GB} = 1.024\text{ MB}$).

A marathon runner maintains a constant pace of $4\text{ minutes and } 15\text{ seconds}$ for each kilometer run.

1. a) How much total time will it take to complete a $15\text{ km}$ race? Express the result in hours, minutes and seconds.
2. b) If the race started at $10:45\text{ h}$, at what time will they cross the finish line?

A communications satellite orbits the Earth and completes exactly one revolution every $103\text{ minutes and } 24\text{ seconds}$. If the satellite began its first orbit on a Monday at $08:15:00$, at what exact time, expressed in hours, minutes and seconds, will it have completed its fifteenth revolution? Also indicate whether it will still be Monday or if the day will have changed.

A broken faucet leaks one drop of water every $2,5\text{ seconds}$. It has been measured that $20\text{ gotas}$ are exactly equivalent to $1\text{ mililitro}$. If the faucet drips non-stop throughout the month of February in a leap year:

1. a) How many liters of water will have been wasted by the end of the month?
2. b) If a cubic meter of water costs $1,85\text{ \$}$, what is the economic value of the lost water?

A wall clock gains $18\text{ seconds}$ every $4\text{ hours}$. If it is set to the exact time on March $1$ at $12:00$ noon, what time will this clock show when the real time is exactly $12:00$ noon on March $11$? Give the answer in hours, minutes and seconds.

Two angles are complementary if their sum is $90^\circ$. Calculate the angle missing from each of the following to be complementary:

1. a) $\hat{A} = 35^\circ$
2. b) $\hat{B} = 52^\circ$ $30'$
3. c) $\hat{C} = 89^\circ$

A graphic designer is drawing the logo for a new app. The logo uses a main angle of $125^\circ \ 40' \ 12''$.

1. a) The designer needs to divide that angle into three equal parts to create a fan effect. What will be the size of each of the three resulting angles?
2. b) What is the supplementary angle of the original angle?

Calculate the size of the interior angle of a regular polygon with $9$ sides (nonagon). Express the result in complex form (degrees, minutes and seconds). Remember that the sum of the interior angles of a polygon with $n$ sides is calculated using the formula: $S = (n - 2) \cdot 180^\circ$.