Exercises: Volume

For a class project, you have to estimate how much water fits in your backpack. If we assume that your bag has a capacity of $18.500 \text{ cm}^3$, how many liters of water could it hold if it were completely waterproof?

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

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

A famous soft drink brand uses cylindrical cans for its drinks. If the base radius of the can is $3 \text{ cm}$ and its height is $12 \text{ cm}$, calculate the volume of soft 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 floor) 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-based 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, cardboard hats have been made in the shape of a cone. 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 table tennis ball has a radius of $2 \text{ cm}$. Calculate the volume of the table tennis 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 laboratory they have a small aquarium in the shape of a rectangular prism. Its measurements are $40 \text{ in}$ long, $20 \text{ in}$ wide and $25 \text{ in}$ high. If we fill it with water to a height of $20 \text{ in}$, how many gallons of water have we poured in?

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

You have decided to build a gel ant farm for a science project. The container has the shape of a rectangular prism with dimensions of $25\text{ cm}$ long, $10\text{ cm}$ wide and $18\text{ cm}$ high. If the gel is sold in tubs of $0,5\text{ liters}$, 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 base radius is $3\text{ in}$ and the slant height measures $5\text{ in}$. Calculate the volume of perfume that fits in the bottle expressed in cubic inches. Hint: you will first need to find the height of the cone using the Pythagorean theorem.

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

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

For a video game event, a trophy has been designed consisting of a regular square-based bronze pyramid. The side of the square base measures $12\text{ cm}$ and the lateral edge (the one joining the vertex to a corner of the base) measures $10\text{ cm}$. Calculate the volume of bronze needed to make one trophy.

Cavalieri's principle states that two solids with the same height and whose cross-sections at the same distance from the base have the same area, possess the same volume. Compare a regular hexagonal prism with a height of $20\text{ cm}$ and a base area of $30\text{ cm}^2$ with a cylinder of the same height and the same base area. If we pour $0,5\text{ liters}$ of water into each one, which 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 $1200\text{ ft}^3$. If we know that the length of the pool is $50\text{ meters}$ and the width is $12\text{ meters}$, calculate the depth (height) of the pool. If the city council decides to fill it only to $4/5$ of its capacity for safety, how many gallons of water will there be in the pool?

A designer has created a trophy for an eSports tournament with a complex geometric shape. The trophy consists of a regular square-based wooden prism on which a regular square-based metal pyramid rests. The edge of the common base measures $12\text{ cm}$. The height of the prism is $20\text{ cm}$, and we know that the lateral edge of the pyramid measures $10\text{ cm}$. Calculate the total volume of the trophy in cubic decimeters ($dm^3$).

An industrial steel part has the shape of a rectangular prism with dimensions $10\text{ in} \times 8\text{ in} \times 6\text{ in}$. To lighten its weight, a cylindrical hole has been made through the part from top to bottom through its largest face. The radius of the cylinder is $2\text{ in}$. Knowing that the density of steel is $7,85\text{ oz/in}^3$, calculate the total mass of the part in pounds. (Take $\pi \approx 3,14$).

In the physics laboratory, the students completely submerge a lead sphere in a cylindrical container half-filled with water. The radius of the container is $10\text{ in}$. When the sphere is introduced, the water level rises exactly $4,5\text{ in}$ without overflowing. What is the radius of the lead sphere? Express the result in inches to two decimal places.

A water tank has the shape of a frustum of a regular square-based pyramid. The edges of the bases measure $4\text{ ft}$ and $2\text{ ft}$ respectively. The height of the tank is $3\text{ ft}$. If the tank is filled to three-quarters of its height (that is, the water reaches $2,25\text{ ft}$ from the base), how many gallons of water does it contain? (Note: You will have to use similarity to find the side of the water's surface).

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

An ice cream manufacturer wants to compare two containers. The first has the shape of 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 the same height, but inside it contains a chocolate hemisphere that occupies 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 the vanilla ice cream? Justify your answer by carrying out the calculations.

A concrete ramp is to be built for a skatepark. The ramp has the shape of a right prism whose base is a right trapezoid. The bases of the trapezoid measure $3\text{ ft}$ and $1\text{ ft}$, and its height (length of the base of the ramp on the ground) is $5\text{ ft}$. If the width of the ramp is $4\text{ ft}$, calculate how many cubic feet of concrete are needed and the total cost if the price is $95\text{ \$/ft}^3$.

A perfume company sells its star fragrance in a bottle consisting of a glass cylinder with a diameter of $4\text{ in}$ and a height of $10\text{ in}$, topped by a cone-shaped cap with the same base and a height of $3\text{ in}$. If the bottle is only filled to $85\%$ of its total capacity to leave space for air, how many fluid ounces ($ml$) of perfume does each bottle contain? (Take $\pi \approx 3,1416$).