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 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?

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 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 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.

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.

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 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 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 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 ($hl$).

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 $\pi \approx 3.14$).

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).