Exercises: Volume of solid shapes

Two Rubik's cubes have different sizes but are geometrically similar. The edge of the small cube measures $2 \text{ cm}$ and the edge of the large cube measures $6 \text{ cm}$. If the volume of the small cube is $8 \text{ cm}^3$, calculate the volume of the large cube using the scale factor.

Two similar steel spheres have been manufactured. The volume of the first is $10 \text{ cm}^3$ and the volume of the second is $80 \text{ cm}^3$. Calculate the scale factor $k$ between the radius of the second sphere and that of the first.

An architect designs the model of a skyscraper to a scale of $1:100$. If the volume of the model in the studio is $5000 \text{ cm}^3$, determine the actual volume that the skyscraper will have, expressed in cubic metres ($m^3$).

In a pizzeria, soft drinks are served in two similar cylindrical glasses. The large glass is exactly twice as tall and twice as wide as the small glass. If the capacity of the small glass is $250 \text{ mL}$, how many millilitres of soft drink does the large glass hold?

A plastic tetrahedron has a volume of $15 \text{ cm}^3$. A toy factory decides to build a similar tetrahedron, but tripling the length of all its edges. Calculate the volume of the new tetrahedron.

The ratio between the total surface areas of two similar cones is $\frac{16}{25}$. Determine the ratio between the volumes of these cones.

A regular hexagonal prism has a volume of $120 \text{ cm}^3$. If we construct a similar prism by halving all its linear dimensions, what will be the volume of the new prism?

In a museum there are two similar marble statues. The volume of the first statue is $27$ times the volume of the second. If the height of the second statue is $10 \text{ cm}$, find the height of the first statue.

An official souvenir shop in A Coruña sells scale replicas of the Torre de Hércules made from the same solid metal alloy. The small model is $10\text{ cm}$ tall and has a mass of $150\text{ g}$. The large model, geometrically similar, is $20\text{ cm}$ tall.

Carlos, a tourist, claims: "Since the height of the large tower is double ($20 : 10 = 2$), its mass will also be double, that is, $300\text{ g}$".

1. a) Identify the error in Carlos's reasoning and explain in your own words why the volume and mass of a three-dimensional object do not double when its lengths double.
2. b) Apply the appropriate scale factor to calculate the actual mass of the large replica.

The trade standards of the Ministry of Agriculture for citrus fruit marketed in Spain classify fruits by their caliber (diameter). Consider two Valencian oranges of the Navelina variety that can be modeled as geometrically similar spheres: a medium-caliber one with a diameter of $7\text{ cm}$ and an extra-caliber one with a diameter of $8,4\text{ cm}$. When squeezing the medium orange, exactly $70\text{ ml}$ of juice was obtained.

1. a) Find the scale factor $k$ between the large orange and the medium one.
2. b) Knowing that the amount of juice is proportional to the total volume of the fruit, calculate how many milliliters of juice will be obtained from the large orange.
3. c) How many whole medium oranges would you need to squeeze to match or exceed the juice provided by a single large orange? Justify your answer based on the relationship between their volumes.

A solid bronze miniature of a famous monument weighs $1,5 \text{ kg}$ and has a height of $20 \text{ cm}$. The real monument, built with exactly the same type of bronze and being mathematically similar, has a height of $4 \text{ m}$. Calculate the weight of the real monument in tonnes.

A cocktail glass is in the shape of a perfect inverted cone. Juice is poured into the glass until it reaches exactly half of the total height of the glass. What fraction of the total capacity of the glass is occupied by the juice?

The figure illustrates the glass and the level of juice:

A spherical hot air balloon is heated, causing its radius to increase by $20\%$. Assuming that it maintains its perfectly spherical shape at all times, by what percentage does its volume increase compared to the initial volume?

A tinned food brand sells soup in two similar cylindrical tins. The small tin has a total surface area of $150 \text{ cm}^2$ and a capacity of $400 \text{ mL}$. If the large tin has a total surface area of $337,5 \text{ cm}^2$, calculate the capacity in millilitres of the large tin.

Two fish of the same species have geometrically similar shapes and lengths of $15 \text{ cm}$ and $25 \text{ cm}$. Knowing that the daily amount of food they need is directly proportional to their volume (or body mass), and that the small fish consumes $27 \text{ g}$ of food per day, how many grams per day does the large fish need?

A water tank belonging to an irrigation community is in the shape of a cuboid and has a capacity of $8000 \text{ L}$. It is decided to build a new tank similar to the first one, but designed so that the area of its base is $4$ times larger than the area of the base of the original tank. Calculate the capacity in litres of the new tank.

Two irregular polyhedra are similar. The volume of the smaller polyhedron is $64 \text{ cm}^3$ and that of the larger is $343 \text{ cm}^3$. If the total surface area of the smaller polyhedron is $48 \text{ cm}^2$, find the total surface area of the larger polyhedron.

In the Spanish wine sector, the traditional Bordeaux barrel (used in wineries of the D.O. Ca. Rioja) has a length of $95\text{ cm}$ and an approved capacity of $225\text{ liters}$. A master cooper craftsman has made a perfectly similar miniature replica with a length of $19\text{ cm}$ for tastings and decoration.

1. a) Estimate: Without using a calculator and before performing numerical operations, choose in a reasoned way which of these three options you think will represent the capacity of the small barrel: around $45\text{ liters}$, around $10\text{ liters}$, or less than $2\text{ liters}$. Justify your intuition.
2. b) Calculate: Obtain the linear scale factor $k$ (of reduction) and calculate with mathematical precision the capacity in liters of the miniature barrel.
3. c) Check and assess: Compare the exact result with your initial estimate. Why is intuition usually misleading when estimating how the volume of a three-dimensional body changes when reducing its dimensions to scale?

On an irrigated farm in Castilla-La Mancha, a water tank shaped like an inverted cone (with the apex resting on the ground) is used. The total height of the tank is $6\text{ m}$ and its maximum capacity is $54\text{ m}^3$. After a few days of irrigation, the water remaining stored in the conical bottom reaches a height of $4\text{ m}$ from the apex. To calculate the available volume of water, two technicians debate which strategy to follow:

• Method A: Solve for the top radius of the cone using the classic formula $V = \frac{1}{3} \cdot \pi \cdot r^2 \cdot h$, calculate the radius of the water at a height of $4\text{ m}$ using similar triangles, and use the cone formula again.

• Method B: Consider that the cone of water is similar to the full tank with ratio $k = \frac{4}{6} = \frac{2}{3}$ and directly apply the relationship of similar volumes $V_{\text{water}} = k^3 \cdot V_{\text{total}}$.

1. a) Solve the problem by calculating the exact volume of water using Method B.
2. b) Carry out Method A (rounding intermediate radii to two decimal places) and calculate the approximate volume of water.
3. c) Compare both methods and justify which one is preferable in this case in terms of speed of calculation and precision (absence of rounding errors).

A square-based pyramid has a height of $12 \text{ cm}$ and its base edge measures $6 \text{ cm}$. This pyramid is cut by a plane parallel to the base to form a frustum of a pyramid whose top base has an edge of $2 \text{ cm}$. Using the similarity between the original pyramid and the smaller top pyramid, calculate the volume of the resulting frustum of a pyramid.

A conical funnel of height $H$ is completely full of oil. All the oil is allowed to drip into a conical container similar to the funnel but of height $2H$, which is initially empty. Once the funnel has emptied completely, what fraction of the height of the large container will be covered by the oil?

Two similar solids have volumes $V$ and $2V$ respectively. If the surface area of the smaller solid is $A$, express the surface area of the larger solid in terms of $A$. Simplify the expression using surds.

Planet Earth has a diameter that is approximately $3,6$ times greater than the diameter of the Moon. Assuming that both celestial bodies are perfect spheres, and knowing that the average density of Earth is $5,5 \text{ g/cm}^3$ while that of the Moon is $3,3 \text{ g/cm}^3$, calculate the ratio between the mass of Earth and the mass of the Moon (remember that $Masa = Volumen \times Densidad$).

An ice sculpture in the shape of a complex polyhedron melts uniformly under the sun such that at all times it maintains a shape similar to the original. If after 2 hours it is observed that its surface area has decreased by $36\%$, what percentage of its original volume has melted up to that point?

A metallurgical factory manufactures solid steel spheres of radius $R = 10 \text{ cm}$. To cut costs, they wish to manufacture a new similar sphere using exactly half the amount of steel as the original. What should the radius $r$ of the new sphere be? Give the exact result and an approximation rounded to two decimal places.

An agricultural silo consists of a cylinder of height $h$ and radius $r$, topped by a hemisphere of the same radius $r$. A scale model is built such that the total volume of the model is $\frac{1}{27}$ of the volume of the real silo. Knowing that in the real silo the height of the cylinder is triple its radius ($h = 3r$), determine the ratio between the total surface area of the model and the total surface area of the real silo.

A glass is in the shape of an inverted frustum of a cone. The radius of the top base is $4 \text{ cm}$, the radius of the bottom base is $2 \text{ cm}$, and the height of the glass is $10 \text{ cm}$. Virtually extending the side walls of the glass forms a complete cone. Using similar triangles in the cross-section to find the height of the removed top cone, calculate the exact volume of the glass.

A master chocolatier from Astorga (León) designs two similar solid figures (a children's figure and a family figure) molded from the same chocolate for the Easter campaign. The family figure has a height that is three times that of the children's figure (length scale factor $k = 3$). Furthermore, due to cost and packaging constraints, the mass of chocolate of the large figure must exceed that of the small one by exactly $1040\text{ g}$.

1. a) Write an algebraic equation involving the unknown $m$ (mass in grams of the children's figure) and the volume ratio, reflecting that the difference in mass between both pieces is $1040\text{ g}$.
2. b) Solve the equation to determine the exact mass of the children's figure and the mass of the family figure.
3. c) If, looking ahead to the following year, the chocolatier decided to modify the mold so that the large figure weighed $4000\text{ g}$ and the small one $500\text{ g}$, what scale factor $k$ should he order to be applied to the heights of the new molds? Set up the corresponding equation from the volumes and find the exact value of $k$.

According to the data sheet from the Museo Arqueológico Nacional (MAN) in Madrid, the original Iberian bust of the Dama de Elche is carved in limestone, has a height of $56\text{ cm}$, and a mass of $65\text{ kg}$. A city council commissions a sculpture workshop to make a monumental replica, geometrically similar and made of the same type of limestone, to install on a pedestal at a roundabout, with a projected height of $2,80\text{ m}$.

1. a) Calculate the scale factor $k$ between the monumental sculpture and the original piece.
2. b) The boom truck hired for the transport has a maximum load limit of $7,5\text{ tons}$ ($7500\text{ kg}$). Deduce in a reasoned way the mass in kilograms of the monumental replica and indicate with justification whether the truck will be able to transport it safely.
3. c) If the outer surface area of the original bust is approximately $0,4\text{ m}^2$ and a protective water-repellent treatment costing $35\text{ €}$ per square meter of surface is to be applied to the monumental sculpture, calculate the total cost of said treatment.
4. d) If, due to crane limitations, the city council required that the total mass of the statue not exceed $4160\text{ kg}$, determine what the maximum height (in meters) the replica could have while maintaining similarity with the original.

Topic challenge. Imagine that a brand hires you to redesign commercial packaging. How does scale influence material usage and capacity?

1. Real measurement: Choose a prism-shaped box at home (cereal, cookies). Measure its edges, calculate the area of its rectangular faces, and its volume. Draw a triangle on one face and calculate its perimeter.

2. Indirect measurement: Use triangle similarity (the mirror reflection method or the shadow method) to calculate the actual height of the shelf where you would store the box.

3. Scaling in a spreadsheet: Design two similar versions in a spreadsheet by applying the scale factors $k = 0,5$ and $k = 1,5$. Automatically compute the new dimensions ($k$), the total cardboard surface area ($k^2$), and the resulting volume ($k^3$).

4. Reasoned proposal: Write a short report justifying whether it is profitable for the company to manufacture the large package (ratio between cardboard used and capacity gained) and whether it fits on the measured shelf.

Assessment criteria:

• Accuracy in indirect measurement using triangle similarity.

• Rigorous use of scale factors $k$, $k^2$, and $k^3$ for lengths, areas, and volumes.

• Proficiency in using the spreadsheet and critical justification of conclusions.