Exercises: Volume measurement

Convert to the specified units:

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

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

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

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

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

Complete in your notebook:

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

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

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

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.

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.

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 decimeters and in cubic centimeters.

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

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

During a storm, $45$ liters per square meter of rain fell.

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

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

c) How many cubic meters 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 liters of water fit in the pool?

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

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 liters 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 centimeters (take $\pi \approx 3.14$).

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

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

b) If Marta 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?

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

a) Express this amount in liters.

b) If the consumption is split equally over the $7$ days of the week, how many liters are consumed per day?

c) Express the daily consumption in compound units.

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 $1.5$ metric tons. 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 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 hectometers of water does it contain?

b) How many liters is that? Express the result in scientific notation.

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

When converting $2.5\ \text{m}^3$ to cubic centimeters, Pablo 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 liters are $2.5\ \text{m}^3$?

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 $45$ liters per minute.

a) How many liters 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 $1$ liter.

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

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 $1\,000$.

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 decimeters.

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?