Exercises: Volume

For a class project, you have to estimate how much water fits into your school backpack. Assuming your backpack has a capacity of $18,500 \text{ cm}^3$, how many litres of water could it hold if it were completely waterproof?

Remember that $1 \text{ dm}^3$ is equivalent to $1 \text{ litre}$.

You have bought a new smartphone and want to calculate the volume of its box to find out how much space it will take up on your shelf. The box is a cuboid (rectangular prism) with the following dimensions: length of $16 \text{ cm}$, width of $9 \text{ cm}$ and height of $5 \text{ cm}$. Calculate its volume in $\text{cm}^3$.

A famous soft drink brand uses cylindrical cans for its drinks. If the radius of the base of the can is $3 \text{ cm}$ and its height is $12 \text{ cm}$, calculate the volume of drink that the can can hold. (Use $\pi \approx 3.14$).

You are camping with your friends and you have a tent in the shape of a right triangular prism. The base of the tent (the ground) is a triangle with a base of $2 \text{ m}$ and a height of $1.5 \text{ m}$. If the tent has a length (depth) of $3 \text{ m}$, what is the volume of air inside the tent?

You have a glass paperweight in the shape of a regular square pyramid. The area of its square base is $25 \text{ cm}^2$ and the height of the pyramid is $9 \text{ cm}$. Calculate the volume of glass needed to make this paperweight.

For a birthday party, party hats in the shape of a cone have been made out of card. If each hat has a radius of $8 \text{ cm}$ and a height of $15 \text{ cm}$, what volume of air is trapped inside the hat when someone is wearing it? (Use $\pi \approx 3.14$).

A ping-pong ball has a radius of $2 \text{ cm}$. Calculate the volume of the ping-pong ball in $\text{cm}^3$. (Use the formula for the volume of a sphere $V = \frac{4}{3} \pi r^3$ and take $\pi \approx 3.14$).

In the science lab, they have a small aquarium in the shape of a cuboid. Its dimensions are $40 \text{ cm}$ long, $20 \text{ cm}$ wide and $25 \text{ cm}$ high. If we fill it with water up to a height of $20 \text{ cm}$, how many litres of water have we poured in?

Aoife wants to calculate the volume of a cylindrical scented candle. The candle measures $10\text{ cm}$ in diameter and $12\text{ cm}$ in height. In her notebook, she has written the following solution:

1. a) Identify and explain in detail what conceptual error Aoife made in her approach.
2. b) Correct her working and calculate the actual, correct volume of wax contained in the candle (take $\pi \approx 3.14$).
3. c) If Aoife were to use her incorrect result to order wax from the factory, how many times more material than necessary would she have ordered? Justify your answer using the volume formula without needing to perform operations with decimals again.

A ball manufacturer produces size 7 basketballs, which are spherical in shape with a radius of $12\text{ cm}$. For distribution, each inflated ball is packed into an individual cubic box into which it fits exactly (with edges of $24\text{ cm}$).

1. a) Before calculating the volume of the sphere, mentally estimate the volume of the cubic box in liters (remember that $1\text{ dm}^3 = 1\text{ liter} = 1,000\text{ cm}^3$). Based on that figure, make a reasoned estimate: do you think the ball will occupy more or less than half the volume of the box?
2. b) Calculate the exact volume of the spherical ball by applying the corresponding formula (use $\pi \approx 3.14$) and express it in liters, rounding to two decimal places.
3. c) Check whether your initial estimate was correct by calculating the exact percentage of the box's volume that the ball actually occupies. What do you think about the air space that goes unused during transport?

A well-known milk-shake brand has designed a new cylindrical container for its summer limited edition. The can has a diameter of $8\text{ cm}$ and a height of $15\text{ cm}$. If the manufacturer indicates that the milk shake occupies $90\%$ of the container's total capacity to leave space for air, calculate how many millilitres ($ml$) of product each can contains.

You have decided to build a gel ant farm for a science project. The container is shaped like a cuboid (rectangular prism) with dimensions $25\text{ cm}$ long, $10\text{ cm}$ wide and $18\text{ cm}$ high. If the gel is sold in tubs of $0.5\text{ litres}$, how many full tubs will you need to buy to fill the ant farm to a height of $15\text{ cm}$?

A perfume company launches a fragrance in a bottle shaped like a right cone. The radius of the base is $3\text{ cm}$ and the slant height is $5\text{ cm}$. Calculate the volume of perfume that fits in the bottle, expressed in cubic centimetres. Hint: first you will need to find the height of the cone using Pythagoras' theorem.

In a science museum, there is an interactive display featuring a glass sphere filled with a noble gas. If the diameter of the sphere is $60\text{ cm}$, what is the volume of gas inside it? Express the answer in cubic metres ($m^3$).

A grain storage silo has a composite shape: a cylindrical main body $10\text{ m}$ high and a hemispherical dome on top. If the radius of the base is $4\text{ m}$, calculate the total capacity of the silo in hectolitres ($hl$).

For a video game event, a trophy has been designed consisting of a square-based right pyramid made of bronze. The side length of the square base is $12\text{ cm}$ and the slant edge (joining the apex to a corner of the base) measures $10\text{ cm}$. Calculate the volume of bronze needed to manufacture one trophy.

Cavalieri's principle states that two solids of the same height whose cross-sections at the same distance from the base have the same area also have the same volume. Compare a regular hexagonal prism with a height of $20\text{ cm}$ and a base area of $30\text{ cm}^2$ to a cylinder with the same height and the same base area. If we pour $0.5\text{ litres}$ of water into each, which one will fill up first? Justify your answer by calculating the volume of both.

A municipal swimming pool in the shape of a rectangular prism has a capacity of $1200\text{ m}^3$. If the length of the pool is $50\text{ metres}$ and the width is $12\text{ metres}$, calculate the depth (height) of the pool. If the council decides to fill it to only $4/5$ of its capacity for safety reasons, how many litres of water will be in the pool?

An artisan master ice cream maker from Galway wants to commission the design of a new conical mold to serve a sorbet. He has determined that the mold must have an exact capacity of $200\text{ ml}$ when filled completely to the brim (remember that $1\text{ ml} = 1\text{ cm}^3$). For reasons of stability on the table, he has also decided that the height of the cone must be exactly four times the radius of its circular base ($h = 4r$).

1. a) Write the formula for the volume of a cone substituting the height $h$ with the expression $4r$, and simplify it so that it depends solely on the radius $r$.
2. b) Set up the corresponding equation by setting the previous formula equal to the desired volume ($200\text{ cm}^3$) and find the value of the radius $r$ and the height $h$. Round the dimensions to the nearest tenth of a centimeter (take $\pi \approx 3.14$).
3. c) Carry out the check: calculate the volume of the cone using the dimensions you obtained and verify that the result is close to the required $200\text{ ml}$ (the small difference is due to rounding).

According to the records of a Met Éireann meteorological station in County Galway, during an episode of torrential rain, an accumulation of $75\text{ l/m}^2$ (liters per square meter) was recorded. An agricultural nursery in the area has a flat polycarbonate roof measuring $20\text{ m}$ long by $8\text{ m}$ wide that collects all the fallen water and channels it entirely into a cylindrical cistern with an inner diameter of $3\text{ m}$ and a depth of $4\text{ m}$, which was completely empty before the storm.

1. a) Model the situation and calculate the total volume of rainwater (both in liters and in cubic meters) collected by the greenhouse roof.
2. b) Determine the height in meters that the water will reach inside the cistern after the episode ends (use $\pi \approx 3.14$ and round to two decimal places).
3. c) Interpret the result obtained: is there any risk that the cistern overflowed? If not, how many additional liters would have been needed to fill it completely?

A designer has created a trophy for an eSports tournament with a complex geometric shape. The trophy consists of a square-based right prism made of wood, topped by a square-based right pyramid made of metal. The edge of the shared base measures $12\text{ cm}$. The height of the prism is $20\text{ cm}$, and the slant edge of the pyramid measures $10\text{ cm}$. Calculate the total volume of the trophy in cubic decimetres ($dm^3$).

An industrial steel part has the shape of a cuboid measuring $10\text{ cm} \times 8\text{ cm} \times 6\text{ cm}$. To lighten its weight, a cylindrical hole has been drilled straight through from top to bottom through its largest face. The radius of the cylinder is $2\text{ cm}$. Given that the density of steel is $7.85\text{ g/cm}^3$, calculate the total mass of the part in kilograms. (Take $\pi \approx 3.14$).

In physics lab, students completely submerge a lead sphere in a cylindrical container half full of water. The radius of the container is $10\text{ cm}$. When the sphere is inserted, the water level rises by exactly $4.5\text{ cm}$ without overflowing. What is the radius of the lead sphere? Express the answer in centimetres to two decimal places.

A water tank is shaped like a frustum of a square-based pyramid. The edges of the bases measure $4\text{ m}$ and $2\text{ m}$ respectively. The height of the tank is $3\text{ m}$. If the tank is filled to three quarters of its height (that is, the water reaches $2.25\text{ m}$ from the base), how many litres of water does it contain? (Note: You will need to use similarity to find the side length of the surface of the water).

Calculate the volume of a regular hexagonal pyramid whose slant edge measures $13\text{ cm}$ and whose base edge measures $10\text{ cm}$. Remember that in a regular hexagon, the radius of the circumscribed circle is equal to the side length. Express the result in $cm^3$, rounding to the nearest whole unit.

An ice cream manufacturer wants to compare two containers. The first is shaped like a cone with a radius of $6\text{ cm}$ and a height of $15\text{ cm}$. The second is a cylinder with the same radius and height, but contains a chocolate hemisphere filling the bottom part. If the radius of the hemisphere is also $6\text{ cm}$, which of the two containers has a greater volume available for vanilla ice cream? Give reasons for your answer by showing your calculations.

A concrete ramp is to be built for a skate park. The ramp is shaped like a right prism whose base is a right-angled trapezium. The parallel sides (bases) of the trapezium measure $3\text{ m}$ and $1\text{ m}$, and its height (length of the base of the ramp on the ground) is $5\text{ m}$. If the width of the ramp is $4\text{ m}$, calculate how many cubic metres of concrete are needed and the total cost if the price is $95\text{ €/m}^3$.

A perfume company sells its star fragrance in a bottle consisting of a glass cylinder $4\text{ cm}$ in diameter and $10\text{ cm}$ in height, topped by a cone-shaped lid with the same base and $3\text{ cm}$ in height. If the bottle is filled to only $85\%$ of its total capacity to leave room for air, how many millilitres ($ml$) of perfume does each bottle contain? (Take $\pi \approx 3.1416$).

A machining workshop needs to manufacture solid bronze parts in the shape of a right prism with a regular hexagonal base. The height of the part is $15\text{ cm}$, the side length of the hexagonal base is $8\text{ cm}$, and the apothem of the base is approximately $6.93\text{ cm}$. Two interns propose two methods to calculate the volume of metal:

• Method 1 (Cian): Calculates the area of the hexagon using the standard formula for a regular polygon ($A = \frac{perimeter \times apothem}{2}$) and multiplies it by the height of the prism.

• Method 2 (Aoife): Mentally divides the hexagonal prism into $6$ identical right triangular prisms whose base is a triangle with a base of $8\text{ cm}$ and a height of $6.93\text{ cm}$, calculates the volume of one of them, and multiplies it by $6$.

1. a) Apply both methods numerically and check whether they lead to the exact same result.
2. b) Justify algebraically why both methods are mathematically equivalent by analyzing their general formulas.
3. c) In the final design, it is decided to drill through the part from top to bottom with a central cylindrical hole $4\text{ cm}$ in diameter to reduce weight. Which of the two conceptual methods is more cleanly adapted to calculate the volume of the remaining metal? Explain your choice and calculate the final volume of the perforated part (take $\pi \approx 3.14$).

In a cooperative in Munster, a metal silo is planned for grain storage. The main body is a vertical cylinder with a base radius of $4\text{ m}$ and a height of $12\text{ m}$, topped at the upper part by a conical closing hopper whose radius matches that of the cylinder and whose slant height is $5\text{ m}$.

1. a) Mentally visualize the cross-section of the conical roof, identify the right triangle formed by its elements, and use the Pythagorean theorem to deduce the exact height of the upper cone.
2. b) Knowing that the bulk density of dry wheat is $0.8\text{ t/m}^3$ ($0.8$ metric tons per cubic meter), calculate the maximum mass of wheat in metric tons that the fully filled silo can hold (cylinder and cone together). Take $\pi \approx 3.14$.
3. c) The manager of the silo claims: "Since the height of the cone is one fourth the height of the cylinder, the cone must contribute approximately $25\%$ of the total volume of the silo." Analyze this claim mathematically by calculating the actual percentage that the cone's volume represents relative to the total volume, and explain with geometric arguments why the manager's reasoning is incorrect.