Measuring volume

Convert to the specified units:

a) $3\ \text{m}^3$ to cubic decimetres.

b) $4\,500\ \text{dm}^3$ to cubic metres.

c) $0.75\ \text{dm}^3$ to cubic centimetres.

In which unit would you express the volume of each of these objects? Choose between $\text{cm}^3$, $\text{dm}^3$, $\text{m}^3$ and $\text{hm}^3$, and justify your choice:

a) A sugar cube.

b) A backpack.

c) The water in a reservoir.

d) A municipal swimming pool.

Look at the conversion step diagram for units of volume in the figure.

a) By what number do you multiply when going down a step? And by what number do you divide when going up?

b) Using the diagram, express $2.4\ \text{m}^3$ in cubic decimetres and in cubic centimetres.

Nerea has to convert $3.4\text{ m}^3$ to liters to find out how much water an inflatable paddling pool needs. In her notebook, she wrote the following procedure:

"Since $1\text{ dm}^3$ is equivalent to $1\text{ liter}$, I first convert to cubic decimeters: $3.4\text{ m}^3 = 340\text{ dm}^3$, because going from meters to decimeters is going down one step, and in the table you add a zero or move the decimal point one place for each dimension of the plane, that is, two zeros. Therefore, it holds $340\text{ liters}$".

a) Identify the mistake Nerea made when converting the volume units.

b) Using the idea of a cube with an edge of $1\text{ m}$ made up of little cubes of $1\text{ dm}$, explain why going down one volume step means multiplying by $1\,000$ and not by $10$ or $100$.

c) Correct Nerea's calculations and determine the actual capacity of the pool in liters.

An irrigation community consumes $1\ \text{dam}^3\ 260\ \text{m}^3$ of water in one week.

a) Express this amount in litres.

b) If consumption is evenly distributed among the $7$ days of the week, how many litres are consumed per day?

c) Express daily consumption in cubic metres.

A reservoir has a capacity of $0.45\ \text{km}^3$. After a year of drought, it is at $62\%$ of its capacity.

a) How many cubic hectometres of water does it contain?

b) How many litres is that? Express the result in standard form.

c) How many cubic hectometres are needed to fill it completely?

When converting $2.5\ \text{m}^3$ to cubic centimetres, Mihai writes in his notebook: $2.5\ \text{m}^3 = 2.5 \times 1\,000 = 2\,500\ \text{cm}^3$.

a) Explain the mistake he has made.

b) Carry out the correct conversion.

c) How many litres are $2.5\ \text{m}^3$?

Complete in your notebook:

a) $75\ \text{l} =$ ____ $\text{dm}^3$

b) $2\,500\ \text{cm}^3 =$ ____ litres

c) $0.5\ \text{m}^3 =$ ____ litres

Out of the three proposed values for each volume, only one is sensible. Choose it and justify your answer:

a) A soft drink can: $3.3\ \text{cm}^3$, $330\ \text{cm}^3$ or $3.3\ \text{dm}^3$.

b) A washing machine: $55\ \text{dm}^3$, $250\ \text{dm}^3$ or $2.5\ \text{m}^3$.

c) A glass of water: $25$ ml, $250$ ml or $2.5$ l.

A water tank has a volume of $2.5\ \text{m}^3$. How many litres of water fit in it? How many $5$ litre containers could be filled?

In the science lab there is a cuboid-shaped aquarium whose interior dimensions are $60\text{ cm}$ long, $30\text{ cm}$ wide, and $40\text{ cm}$ high.

a) Before doing detailed calculations, make a quick estimate: knowing that $1\text{ dm}^3 = 1\text{ liter}$, do you think filling it will take more than $100$ bottles of $1.5\text{ liters}$ of water or fewer than $100$? Write down your reasoned prediction.

b) Mentally convert the three dimensions to decimeters and calculate the exact volume of the aquarium in cubic decimeters and its capacity in liters.

c) Calculate exactly how many $1.5\text{-liter}$ bottles are needed to fill it completely and compare this value with your initial estimate. Was your intuition correct?

During a storm, $45$ litres per square metre of rain fell.

a) How many millimetres of water depth is this equivalent to?

b) How many litres fell on a garden of $60\ \text{m}^2$?

c) How many cubic metres is that?

A tank contains $1.2\ \text{m}^3$ of water. How many $750$ ml bottles can be filled with it?

An above-ground swimming pool is $3\ \text{m}$ long, $2\ \text{m}$ wide and $0.75\ \text{m}$ deep.

a) How many litres of water fit in the pool?

b) If it is filled using a hose that pours $15$ litres per minute, how many hours does it take to fill up?

A juice cooperative is designing a cuboid-shaped cardboard carton. To comply with commercial regulations, the package must contain exactly $1\text{ liter}$ of juice and also have an empty space of $50\text{ cm}^3$ at the top to prevent spills when opened.

a) What must be the total interior volume of the package expressed in cubic centimeters ($\text{cm}^3$)?

b) If, for ergonomic reasons, the base of the carton must be a rectangle $7\text{ cm}$ wide by $10\text{ cm}$ long, what exact height must the box have?

c) If the packaging machine cannot handle boxes taller than $12\text{ cm}$, propose new integer interior dimensions for the length, width, and height (in centimeters) that maintain exactly the total required volume.

To fill the irrigation tank for the school garden, water has been collected from three different sources: a drum provides $0.45\text{ m}^3$, another container holds $280\text{ dm}^3$, and a rainwater collector supplies $70\,000\text{ cm}^3$. We want to find out how many $8\text{-liter}$ watering cans can be filled with all the gathered volume. Two students suggest different approaches:

• Clara's method: Convert each of the three amounts beforehand into cubic decimeters (liters), add them together, and divide the total by $8$.

• David's method: Express all amounts in cubic meters with decimal numbers, add them together, convert the final sum to cubic centimeters, and divide by $8\,000\text{ cm}^3$.

a) Carry out the calculations step by step following both methods and verify that they lead to the same solution.

b) Which of the two methods do you consider more efficient and safer to avoid mistakes when shifting decimal points or adding zeros? Justify your choice based on the equivalences between Decimal Metric System units and capacity units.

A garage with a surface area of $45\ \text{m}^2$ has flooded and the water has reached a height of $0.6\ \text{m}$. The water is pumped out with a pump that removes $45$ litres per minute.

a) How many litres of water are in the garage?

b) How long will it take the pump to empty it? Express the result in hours.

State whether these statements are true or false, giving reasons:

a) $1\ \text{dm}^3$ is equivalent to $1$ litre.

b) $1\ \text{m}^3$ is equivalent to $100$ litres.

c) $1\ \text{cm}^3$ of water has a mass of $1$ gram.

d) To convert from $\text{m}^3$ to $\text{cm}^3$, multiply by $1\,000$.

Met Éireann, the Irish Meteorological Service, records an accumulated rainfall of $65\text{ l/m}^2$ in a town during an autumn storm. The roof of the municipal sports center is flat and has a rectangular surface area of $40\text{ m}$ long by $25\text{ m}$ wide. All the water falling on this roof is channeled into a cubic underground cistern with an edge length of $4\text{ m}$ which, before the rain began, already contained $12\,000\text{ liters}$ of water.

a) Reasoned through unit conversions of volume and area, prove why a rainfall of $1\text{ l/m}^2$ is exactly equivalent to a water depth or layer of $1\text{ mm}$ thickness.

b) Mathematically model the situation and calculate the volume of rainwater, in cubic meters, that will fall on the roof during the storm.

c) Determine whether the cistern will overflow or if it will have enough capacity to store all the collected water. Justify your answer by indicating how many liters it will be short of filling up or how many liters will overflow.

Unit Challenge. Geometric audit and sustainability in your pantry.

1. Data collection: Choose two everyday packaging items at home: a polyhedron (such as a carton or cereal box) and a solid of revolution (a cylindrical can or jar). Measure their dimensions in centimeters using a ruler and note the net capacity indicated on the label (in $ml$ or $cl$).

2. Calculation and equivalences: Model both in GeoGebra 3D or record their data in a spreadsheet. Find the total surface area ($cm^2$) and theoretical volume ($cm^3$) of each solid. Convert that volume to liters ($1\text{ dm}^3 = 1\text{ L}$) and compare it with the label value, explaining any discrepancies due to wall thickness or safety clearance.

3. Reasoned proposal: Calculate the $\text{Area}/\text{Volume}$ ratio for both containers. Prepare a short report (one page) proposing the most efficient shape to market $0.5\text{ L}$ of product using the least amount of material possible and minimizing the ecological footprint.

Assessment criteria:

• Precision in real measurements, algebraic calculation of surface areas and volumes, and correct equivalence between units of capacity and volume.

• Rigorous use of digital tools (spreadsheet or GeoGebra) to represent and analyze three-dimensional solids.

• Quality of the final proposal and mathematical justification regarding packaging optimization.

Express in cubic decimetres: $3\ \text{m}^3\ 450\ \text{dm}^3\ 25\ \text{cm}^3$.

Express in compound units: $12\,345\,678\ \text{mm}^3$.

Express in cubic metres and calculate: $2\ \text{dam}^3\ 15\ \text{m}^3 + 840\ \text{m}^3\ 250\ \text{dm}^3$. Express the result in hectolitres as well.

An empty tank weighs $12$ kg and, when full of diesel, weighs $522$ kg. If the density of diesel is $0.85\ \text{kg/dm}^3$, how many litres of diesel does the tank contain?

To measure the volume of a stone, we submerge it in a cylindrical glass containing water, as shown in the figure. The internal diameter of the glass is $8$ cm and the water level rises from $5$ cm to $8$ cm.

a) How much has the water level risen?

b) Calculate the volume of the stone in cubic centimetres (take $\pi \approx 3.14$).

A bathtub is full of water to the brim. Aoife gets into it and $35$ litres of water overflow.

a) What is the volume, in cubic decimetres, of the submerged part of Aoife's body?

b) If Aoife weighs $32$ kg, what is the average density of her body in $\text{kg/dm}^3$? Compare it with the density of water ($1\ \text{kg/dm}^3$) and explain why she can float.

A block of cork has a volume of $25\ \text{dm}^3$ and a density of $0.24\ \text{kg/dm}^3$.

a) What is its mass?

b) Will it float in water? Give reasons for your answer by comparing densities.

c) What volume of cork would be needed to have a mass of $30$ kg?

A concrete block of $1.5\ \text{m} \times 0.8\ \text{m} \times 0.4\ \text{m}$ has a density of $2\,400\ \text{kg/m}^3$.

a) Calculate its volume and its mass.

b) A crane can lift up to $1.5$ tonnes. Can it move the block?

c) Could it move two blocks at the same time? Justify your answer.

Legend has it that Archimedes discovered how to check if the king's crown was made of pure gold by measuring the water it displaced when submerged. In the diagram, a crown weighing $750$ g is submerged in a measuring cylinder containing water.

a) What is the volume of the crown?

b) Calculate its density in $\text{g/cm}^3$.

c) Given that the density of gold is $19.3\ \text{g/cm}^3$ and that of silver is $10.5\ \text{g/cm}^3$, could the crown be solid gold? Give reasons for your answer.

A block of ice of $25\ \text{cm} \times 10\ \text{cm} \times 5\ \text{cm}$ has a density of $0.92\ \text{kg/dm}^3$.

a) Calculate its volume in cubic decimetres.

b) What is its mass?

c) When it melts, the resulting water has a density of $1\ \text{kg/dm}^3$. What volume will it occupy? Does it occupy more or less space than the ice? Why do you think this happens?

To calculate the volume of a solid bronze sculpture with a very irregular shape, it is completely submerged in a tank filled with water to the brim, and all the overflowing liquid is collected in an auxiliary cubic container with an inner edge of $20\text{ cm}$.

a) Knowing that the collected water fills exactly three quarters of the auxiliary container, calculate the volume of the sculpture in cubic decimetres and also express it in litres.

b) When weighing the water displaced by the sculpture on a scale, the instrument shows a mass of $6\text{ kg}$. Explain why this mass measurement confirms that the capacity and volume calculations from the previous section are correct.

c) There are plans to make a $2:1$ scale replica of that sculpture (that is, doubling each of its linear dimensions: length, width, and height). A workshop member states: "Since it is twice as large, submerging it will displace $12\text{ litres}$ of water, which will weigh $12\text{ kg}$". Explain, based on how volume is calculated in three dimensions, why this statement is incorrect and determine the actual volume (in $\text{dm}^3$) and the mass of water that the replica will displace.