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 $2\text{ metres}$ and a height of $5\text{ metres}$. If $1\text{ m}^3$ is equivalent to $1000\text{ litres}$, how many litres 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.

For a town festival, Sara wants to cover the outside of $20$ cardboard party hats with shiny paper. Each hat has a base radius of $9\text{ cm}$ and a height of $12\text{ cm}$ (the hats are hollow and have no base). Her classmate Hugo does the following calculations to find the paper needed for a single hat:

Therefore, Hugo concludes that for the $20$ hats they will need exactly $20 \times 339.12 = 6,782.4\text{ cm}^2$ of paper.

1. a) Identify and explain the two conceptual mistakes Hugo has made: one in the surface area formula he used and another by not calculating a key measurement of the cone.
2. b) Correct the calculations by first finding the slant height of the cone using the Pythagorean theorem and determine the actual minimum amount of shiny paper they need for the $20$ hats.

A town council wants to restore and paint a spherical steel sculpture installed on a roundabout, whose diameter measures exactly $4\text{ m}$. The painter in charge has cans of rust-preventive paint where the manufacturer indicates that each can covers $12\text{ m}^2$.

1. a) Mentally estimate, without a calculator and rounding $\pi \approx 3$, the exterior surface area of the sphere and how many cans would be needed at minimum. Explain the steps of your mental calculation.
2. b) Perform the precise calculation using the formula for the surface area of a sphere ($A = 4\pi r^2$) with the calculator's value of $\pi$ (rounding the area to two decimal places).
3. c) Check and compare the actual result with your estimate: would it have been enough to buy the cans estimated with $\pi \approx 3$ or would the painter have run out of paint before finishing? Justify why this difference occurs when rounding $\pi$.

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 millimetres.

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 $6\text{ metres}$. Engineers want to apply a coat of anti-rust paint costing $\text{€}\ 14.50$ per square metre. 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 $\pi \approx 3.14$).

The Irish Coast Guard inspects a cylindrical-conical navigation buoy in the port of Dún Laoghaire. The buoy consists of a lower cylindrical body with a diameter of $1.2\text{ m}$ and a height of $1.5\text{ m}$, topped at its upper part by a conical tip of the same diameter and a height of $0.8\text{ m}$. A special marine paint must be applied to the entire visible exterior surface: the lateral wall of the cylinder, the submerged bottom flat circular base, and the lateral surface of the cone (the circular joint between the cylinder and the cone is internal and is not painted).

1. a) Calculate the slant height of the conical tip.
2. b) Calculate the total exterior area that must be painted (express the result rounded to two decimal places).
3. c) Each can of marine paint covers a maximum of $4.5\text{ m}^2$ per coat. If technical regulations require applying two full coats and the cans are only sold as whole units at $28.50\text{ €}$ each, how many cans must be purchased and what will be the final cost of the material?

A canned vegetable company in Cork designs a new cylindrical tin can with a top and bottom lid. Due to strength and stacking requirements, it is established that the height of the cylinder must be exactly three times the radius of its base ($h = 3r$). Furthermore, to fit the manufacturing budget, the total area of sheet metal used for each can must be exactly $128\pi\text{ cm}^2$ (about $402.12\text{ cm}^2$).

1. a) Write the algebraic expression for the total area of the cylinder solely in terms of the radius $r$, set up the corresponding equation, and find the exact value of the radius and the height of the container.
2. b) Calculate what percentage of the can's total area corresponds to its lateral surface and what percentage to the two flat bases.
3. c) Reason whether these percentages would change if you decided to double both the radius and the height of the container while maintaining the proportion $h = 3r$.

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 flat piece, 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 $€45$ per square metre of exterior surface and sealing the joint between the hemisphere and the cylinder costs $€15$ per linear metre of circumference, calculate the total budget for the exterior surface materials of the tank (including the flat 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?

To manufacture the lampshade of a designer lamp shaped like a frustum of a cone open at both bases, a lighting workshop in Cork starts with the following measurements: top radius $r = 10\text{ cm}$, bottom radius $R = 25\text{ cm}$, and vertical height of the frustum $h = 20\text{ cm}$. Two members of the technical team, Aoife and Cian, propose different methods to calculate the square metres of fabric needed (lateral area):

• Aoife's Method: Directly calculate the slant height of the frustum using the right-angled triangle formed by the height and the difference between the radii, and apply the formula for the lateral area of a frustum: $A_L = \pi \times (R + r) \times g$.

• Cian's Method: Extend the frustum to complete the original cone. Calculate the slant height of the complete cone and that of the fictitious upper cone using similar triangles, and obtain the lateral area by subtracting both conical areas: $A_L = A_{L,\text{large}} - A_{L,\text{small}}$.

1. a) Apply Aoife's Method to obtain the exact value of the lateral area.
2. b) Apply Cian's Method: set up the similarity ratio to find the required slant heights and calculate the lateral area by subtraction.
3. c) Compare both methods: do the results match exactly? Justify which of the two procedures is more efficient for this numerical calculation and explain at which stage of the actual design (for example, when cutting out the flat fabric pattern before curving it) it is essential to know the measurements obtained with Cian's method.

The Centre for Studies in Physics of the Cosmos is planning a protective dome for a new sensor at the Javalambre Observatory (Teruel). The circular concrete base on which the structure will rest has a fixed radius of $4\text{ m}$. The architects are analysing two covers that reach the same maximum height of $4\text{ m}$ above the ground:

• Option A (Hemispherical): A dome shaped like a hemisphere with radius $r = 4\text{ m}$.

• Option B (Conical): A conical cover with base radius $r = 4\text{ m}$ and apex height $h = 4\text{ m}$.

1. a) Calculate the exterior surface area exposed to the weather for each of the two options (excluding the circular floor). Express the results in terms of $\pi$ and with decimal approximation to the hundredths.
2. b) Prove algebraically that the ratio between the area of the conical cover and the area of the hemispherical dome is a constant value independent of the radius, provided that the height of the cone equals its radius ($h = r$). What is that exact ratio expressed with radicals?
3. c) If the exterior insulating coating costs $130\text{ \euro/m}^2$, calculate the exact monetary savings of choosing the option with the smaller exterior surface area. Considering the high-mountain climate with heavy snowfalls, what additional physical advantage does the more economical geometric option offer regarding the clearing of the weight of accumulated snow?