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 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.

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.

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 $\pi \approx 3.14$.

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 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 $h = 3r$.

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 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).

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 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 $\text{cm}^2$.

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 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 $\pi \approx 3.14$).

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).

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.

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 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.

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 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 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?