Exercises: Solids of revolution

Imagine you have a rectangular piece of card with a base of $10\text{ cm}$ and a height of $15\text{ cm}$. If you spin it quickly around one of its $15\text{ cm}$ sides (the side marked as the axis of rotation in the figure), what solid of revolution is generated? Draw the resulting solid and state its radius and its height.

You can do it on paper and send a photo with your answer.

A standard drink can has a cylindrical shape. If the radius of its base measures $3.25\text{ cm}$ and its height is $12\text{ cm}$, calculate the area of the paper label that wraps around the side of the can (lateral area). Use $\pi \approx 3.14$.

For a wizard costume, a student has made a cone-shaped hat. If the base radius is $10\text{ cm}$ and the height of the hat is $24\text{ cm}$, use Pythagoras' theorem to calculate the slant height ($g$) of the cone.

A tennis ball has an approximate diameter of $6.7\text{ cm}$. Calculate the total surface area of felt needed to cover the ball, treating it as a perfect sphere. Express your answer in $\text{cm}^2$.

Find the volume of a cylindrical water tank with a radius of $2\text{ meters}$ and a height of $5\text{ meters}$. If $1\text{ m}^3$ is equivalent to $1000\text{ liters}$, how many liters of water can the tank hold?

An ice cream cone has the shape of a cone. The radius of the top opening is $3\text{ cm}$ and its depth (height) is $10\text{ cm}$. What volume of ice cream fits exactly inside the cone? (Assume the ice cream is level with the rim, without overflowing).

If we cut an orange in half, we get two hemispheres. If the radius of the orange is $4\text{ cm}$, calculate the volume of one of these halves.

Look at the following net of a solid of revolution.

1. a) Which solid is it?
2. b) If the radius of the circle is $R = 5\text{ cm}$ and the height of the rectangle is $H = 10\text{ cm}$, calculate the total area of the figure.

A photography studio uses a cone-shaped reflector to focus light. If the reflector has a height of $12\text{ cm}$ and the diameter of its circular opening is $10\text{ cm}$, calculate the total surface area of reflective material required to manufacture it (excluding the base cover).

A brand of vitamin supplements designs its capsules in the shape of a cylinder with two hemispheres at the ends. The total length of the capsule is $22\text{ mm}$ and the diameter of the central cylinder is $8\text{ mm}$. Calculate the total volume of the capsule in cubic millimeters.

A glass tumbler has the shape of a cylinder with an internal radius of $3.5\text{ cm}$. If we pour a soft drink into it that occupies a volume of $250\text{ cm}^3$, what height will the liquid reach inside the tumbler? Round your answer to two decimal places.

Calculate the lateral surface area and volume of a frustum of a cone (such as a popcorn bucket) whose bases have radii of $10\text{ cm}$ and $6\text{ cm}$, knowing that its height is $15\text{ cm}$. Note: You will need to calculate the slant height first using Pythagoras' theorem.

A spherical gas storage tank has an outer diameter of $6\text{ meters}$. Engineers want to apply a coat of anti-rust paint costing 290 pesos per square meter. How much will it cost to paint the total surface area of the tank? (Use $\pi \approx 3.14$).

A solid of revolution is generated by rotating a rectangle with base $4\text{ cm}$ and height $7\text{ cm}$ about one of its longer sides.

1. What geometric shape is obtained?

2. Calculate its total surface area.

3. If it were rotated about the shorter side, would it have the same volume? Justify your answer by performing the calculations.

In an ice cream parlour, a perfectly spherical scoop of ice cream of diameter $6\text{ cm}$ is served on a cone-shaped wafer of height $10\text{ cm}$ and the same diameter as the scoop. If the ice cream melts and stays completely inside the cone, will it overflow or will there be space left? Answer by comparing the volumes of the sphere and the cone.

A courier company uses cylindrical cardboard tubes to send blueprints. If each tube has a lateral area of $942\text{ cm}^2$ and a height of $50\text{ cm}$:

1. Calculate the radius of the base of the tube.

2. Calculate the amount of cardboard needed to make the two circular lids for each tube. (Use $\pi \approx 3.14$).

A lamp designer wants to create a lampshade in the shape of a frustum of a cone. The lampshade has a height of $24 \text{ cm}$, a top diameter of $18 \text{ cm}$ and a bottom diameter of $30 \text{ cm}$. To calculate the cost of the material, he needs to know the lateral surface area. However, to manufacture it from a single sheet of material, he must first determine the height of the original full cone from which the frustum comes. Calculate this total height and the lateral area of the lampshade (without the bases).

An ice cream glass has the shape of an inverted cone with height $12 \text{ cm}$ and a top diameter of $10 \text{ cm}$. It is filled with chocolate milkshake to exactly half its height. One student claims that the glass is "half full", but another says that there is much less than half of the total volume. Calculate what percentage of the total volume of the glass the milkshake actually occupies to check who is right.

A steel water tank is to be built, combining a cylinder of height $4 \text{ m}$ and radius $3 \text{ m}$ with a perfect hemisphere at its top. If the cost of the steel is 900 pesos per square meter of exterior surface and sealing the joint between the hemisphere and the cylinder costs 300 pesos per linear meter of circumference, calculate the total budget for the exterior surface materials of the tank (including the circular bottom base).

A stone rolling pin has a cylindrical shape with a radius of $4 \text{ cm}$ and a length of $25 \text{ cm}$. It is to be stored in a cylindrical box such that the rolling pin fits exactly (tangent to the walls and bases). If it is decided to replace this rolling pin with another that has half the radius but double the length:

1. Will the volume of stone used change? Justify your answer with calculations.

2. Which of the two rolling pins would require a box with a larger total surface area?

Calculate the volume of the solid generated by rotating $360^\circ$ the following right-angled trapezium around side $AB$. Side $AD$ measures $6 \text{ cm}$ (shorter base), side $BC$ measures $10 \text{ cm}$ (longer base) and the height of the trapezium (side $AB$) measures $8 \text{ cm}$. Identify the resulting solid and calculate its total surface area.

Two cities, $P$ and $Q$, lie on the same parallel of latitude at $60^\circ \text{ N}$. City $P$ has a longitude of $15^\circ \text{ E}$ and city $Q$ of $45^\circ \text{ E}$. Knowing that the radius of the Earth is approximately $6371 \text{ km}$:

1. Calculate the radius of the parallel $60^\circ \text{ N}$ (Hint: use trigonometry or recall that in the triangle formed, the radius of the parallel is $R \times \cos(60^\circ)$).

2. Find the minimum distance between the two cities measured along the arc of that parallel.

A perfume bottle has the shape of a sphere of radius $R = 5 \text{ cm}$. The bottle is contained inside a box in the shape of a right cylinder that has the same radius as the sphere and a height equal to the diameter of the sphere.

1. Calculate the difference between the volume of the cylindrical box and the volume of the sphere.

2. Show that the lateral area of the cylinder is equal to the surface area of the sphere.

A spherical lead ball with a diameter of $6 \text{ cm}$ has been completely submerged in a cylindrical container of radius $10 \text{ cm}$ containing water.

1. By how much has the water level in the container risen after introducing the ball?

2. If after this, we melt the lead ball and make a cone with it whose base has the same radius as the original sphere ($3 \text{ cm}$), what height will the cone have?