Measuring Volume
A jug of drinking water holds 20 liters, and the cistern of a house is measured in cubic meters. How many jugs fit in a cistern of $2\ \text{m}^3$? To answer, you need to know how to move between units of volume and units of capacity.
La escalera de las unidades de volumen
Volume is measured in cubic units: $1\ \text{cm}^3$ is the space taken up by a little cube with edges of 1 cm. In length, each step is $\times 10$, but in volume each unit is 1,000 times the next one, because $10 \times 10 \times 10 = 1{,}000$.
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…
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.
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.
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.
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…
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 $2.5\ \text{m}^3$?
Which unit should you choose?
Choose the unit according to the size of the object: $\text{cm}^3$ for a sugar cube or a can, $\text{dm}^3$ for a backpack or a washing machine, $\text{m}^3$ for a swimming pool or a cistern, and $\text{hm}^3$ or $\text{km}^3$ for a dam.
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…
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…
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.
Volume and Capacity
Capacity measures how much liquid fits in a container and is expressed in liters. Volume and capacity are the same idea with different units, and the relationship between them is very simple.
That answers the question from the beginning: 100 jugs fit.
Another unit of capacity that comes up is the hectoliter, where $1\ \text{hL} = 100\ \text{L}$. For example, $2{,}500\ \text{L} = 2{,}500 \div 100 = 25\ \text{hL}$. Careful: don't confuse it with the $\text{hm}^3$, which is a huge unit of volume.
Liters per square meter and millimeters of rain
When the forecast says "30 liters per square meter fell," it means that 30 L fell on every $1\ \text{m}^2$ of ground. A liter is $1\ \text{dm}^3$ and a square meter measures $10\ \text{dm} \times 10\ \text{dm}$, so 1 L spread over $1\ \text{m}^2$ forms a layer of height $\frac{1\ \text{dm}^3}{100\ \text{dm}^2} = 0.01\ \text{dm}$, that is, 1 mm.
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
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
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?
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?
A tank contains $1.2\ \text{m}^3$ of water. How many $750$ ml bottles can be filled with it?
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…
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?
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…
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 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 $45$…
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)…
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 $1\,000$.
Complex Form
Sometimes a volume is written split across several units, like $3\ \text{m}^3\ 450\ \text{dm}^3$. This is complex form: each part must be between 0 and 999 (except the largest).
Very large volumes: scientific notation
A dam holds millions of liters, and writing them out with all their zeros is awkward. Scientific notation writes them as $a \times 10^n$, with $1 \le a < 10$. Since $1\ \text{hm}^3 = 1{,}000{,}000\ \text{m}^3 = 1{,}000{,}000{,}000\ \text{L}$, then $1\ \text{hm}^3 = 10^9\ \text{L}$.
Express in cubic decimeters: $3\ \text{m}^3\ 450\ \text{dm}^3\ 25\ \text{cm}^3$.
Express in cubic decimeters: $3\ \text{m}^3\ 450\ \text{dm}^3\ 25\ \text{cm}^3$.
Express in compound units: $12\,345\,678\ \text{mm}^3$.
Express in compound units: $12\,345\,678\ \text{mm}^3$.
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.
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.
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…
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 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)…
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?
Volume, Capacity, and Mass
Water has a very useful property: 1 liter of water weighs 1 kg, that is, $1\ \text{dm}^3$ of water has a mass of 1 kg and $1\ \text{cm}^3$ has a mass of 1 g.
For other materials we use density: the mass of each unit of volume.
It can also be used to measure irregular objects: when you submerge one in water, the volume it takes up equals the volume of water it displaces. According to legend, this is what Archimedes did with the king's crown.
If the container is a cylindrical glass, the water that rises forms a small cylinder. The volume of a cylinder is the area of its base times its height: $V = \pi r^2 h$. Here the height is how much the water level rises.
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…
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 $\pi \approx 3.14$).
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…
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?
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…
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…
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…
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…
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 decimeters. b) What…
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?