Volume of a geometric solid

An aquarium is in the shape of a cuboid with the following dimensions: $60\text{ cm}$ in length, $30\text{ cm}$ in width and $40\text{ cm}$ in height. Calculate how many litres of water fit in the aquarium if it is completely filled. (Remember that $1\text{ dm}^3 = 1\text{ litre}$).

Met Éireann, the Irish Meteorological Service, has recorded a precipitation of $45\text{ mm}$ (equivalent to $45\text{ L/m}^2$) at a station in the Midlands during an autumn storm. A farmer has an open-topped rectangular tank on his plot to collect rainwater, measuring $4\text{ m}$ long, $2.5\text{ m}$ wide, and $1.2\text{ m}$ deep, initially empty. Right next to it, he has a homemade cylindrical rain gauge with an inside diameter of $10\text{ cm}$ and a height of $30\text{ cm}$, also open and empty.

1. a) Calculate how many litres of water fell directly into the rectangular tank during the storm.
2. b) What height in centimetres did the water accumulated inside the rectangular tank reach? And what height will it have reached inside the cylindrical rain gauge? Justify your answer based on the definition of "millimetres of rain".
3. c) What percentage of the total capacity of the rectangular tank has been filled with this storm?

A package containing a gaming keyboard is in the shape of a cuboid. Its dimensions are $45\text{ cm}$ in length, $15\text{ cm}$ in width and $5\text{ cm}$ in height. Calculate the volume of air inside the box in cubic centimetres.

A cylindrical drink can has a diameter of $6\text{ cm}$ and a height of $12\text{ cm}$. Calculate its volume using the value of $\pi \approx 3.14$.

Calculate the volume of a prism whose base is a right-angled triangle with legs $6\text{ cm}$ and $8\text{ cm}$, knowing that the height of the prism is $15\text{ cm}$.

A piece of cheese is in the shape of a cylinder with a radius of $10\text{ cm}$ and a height of $8\text{ cm}$. If we cut the cheese exactly in half vertically, what will be the volume of each of the two resulting parts? (Use $\pi \approx 3.14$).

A decorative glass pyramid has a square base of side $8\text{ cm}$ and a height of $10\text{ cm}$. Calculate the volume of glass needed to manufacture it.

An ice cream cone has the shape of a cone. The radius of the base is $3\text{ cm}$ and its height is $12\text{ cm}$. Calculate how many cubic centimetres of ice cream fit inside it (without overflowing at the top).

An ice cream cone is to be made whose slant height measures $15 \text{ cm}$ and whose base radius is $6 \text{ cm}$. Calculate the volume of ice cream that fits inside the cone. If a perfect hemisphere of ice cream of the same radius is added on top of the base when served, what will be the total volume of the ice cream?

A water tank is in the shape of a frustum of a cone. The radius of the larger base is $3 \text{ m}$, the radius of the smaller base is $1.5 \text{ m}$ and the height of the tank is $4 \text{ m}$. Calculate the capacity of the tank in cubic metres.

(Hint: you can use the formula for the volume of a frustum of a cone or view it as the difference between two cones).

A perfume bottle has the shape of a regular octahedron with an edge length of $6\text{ cm}$. However, the glass has a thickness of $2\text{ mm}$ on all its faces. The manufacturer claims that the container can hold $80\text{ ml}$ of fragrance ($1\text{ ml} = 1\text{ cm}^3$). Perform the necessary calculations to check whether the claim is true. (Given: The volume of an octahedron with edge length $a$ is $V = \frac{\sqrt{2}}{3} a^3$. You must consider how the thickness affects the edge length of the interior cavity).

The PE teacher stores handball balls in a box. If each ball has a radius of $9\text{ cm}$, calculate the volume of one of them rounding to two decimal places. (Use $\pi \approx 3.14$).

Three metal balls of radius $3 \text{ cm}$ are melted down to create a single larger sphere. Assuming no material is lost in the process, what will be the radius of the resulting new sphere? Round the answer to two decimal places.

A cylindrical glass with an inner diameter of $6 \text{ cm}$ contains water up to a height of $10 \text{ cm}$. An irregularly shaped stone is dropped into it and the water level rises to $12.5 \text{ cm}$ without overflowing. In the figure, the dashed line marks the initial water level and the solid blue line marks the final level.

What is the volume of the stone, expressed in cubic centimeters?

Calculate the volume of a geometric solid formed by a cylinder of height $10 \text{ cm}$ and radius $4 \text{ cm}$, topped by a square pyramid whose base is inscribed in the circle of the cylinder and whose height is $6 \text{ cm}$.

(Hint: the side of the square base of the pyramid is equal to the side of a square inscribed in a circle of radius $4 \text{ cm}$).

A County Council has installed concrete planters in the shape of a regular hexagonal prism. If each side of the base hexagon measures $30 \text{ cm}$ and the height of the planter is $60 \text{ cm}$, calculate the volume of soil (in litres) needed to fill it completely.

(Given: remember that you must first find the apothem of the base using Pythagoras' Theorem).

A grain silo has a composite shape: a central cylindrical section with a height of $12\text{ m}$ and a diameter of $4\text{ m}$, topped at the upper part by a hemisphere and at the lower part by a discharge cone with a height of $3\text{ m}$. Calculate the total capacity of the silo in cubic metres. If the silo is filled to $85\%$ of its capacity, how many tonnes of wheat will it contain if the density of wheat is $750\text{ kg/m}^3$?

A lead sphere with a radius of $10\text{ cm}$ is melted down to manufacture small cylinders with a diameter of $2\text{ cm}$ and a height of $3\text{ cm}$. Assuming that $5\%$ of the material is lost in the smelting process, how many complete cylinders can be made? If each cylinder has a mass of $107\text{ g}$, what is the density of the lead used in $g/cm^3$?

Cavalieri's Principle states that if two solids have the same height and the area of their cross-sections at any height is equal, then they have the same volume. Use this idea to compare the volume of a sphere of radius $R$ with the volume contained between a cylinder of radius $R$ and height $2R$, and a double cone inscribed in it. Show mathematically that the volume of the sphere is indeed $V = \frac{4}{3}\pi R^3$ by first calculating the volume of the cylinder and subtracting that of the two cones.