Exercises: Solids of revolution

Imagine you stick the following plane figures to a toothpick and spin them quickly on it. Identify which solid of revolution is generated in each case:

1. a) A rectangle rotating around one of its sides.
2. b) A right triangle rotating around one of its legs.
3. c) A semicircle rotating around its diameter.

In the following drawing of a cone, identify the names of the elements indicated by the letters $r$, $h$ and $g$. Briefly explain what each of them represents.

A cylindrical soda can has a height of $12$ in and its base radius measures $3$ in. Draw a schematic of its net (the figures you would obtain if you cut the can and laid it flat on a table) and indicate the known measurements on the drawing.

To make a cone-shaped party hat, we need to know its slant height ($g$). If the hat has a height of $8$ in and a base radius of $6$ in, use the Pythagorean theorem to calculate how long the slant height must be.

Calculate the volume of a bowling ball (which is a perfect sphere) knowing that its radius measures $11$ in. Use the formula $V = \frac{4}{3} \cdot \pi \cdot r^3$ and take the value of $\pi \approx 3,14$.

A tennis ball tube has the shape of a cylinder. The base has a diameter of $7$ in and the height of the tube is $22$ in. Calculate the lateral surface area of the tube (excluding the lids).

Hint: Remember that the lateral area of a cylinder is $A_L = 2 \cdot \pi \cdot r \cdot h$.

Identify which of these nets could never form a cylinder and explain why.

If we make a horizontal cut (parallel to the base) through a cylinder and a cone, what plane geometric figure will we obtain in the cross-section of the cut? Would it be the same if the cut were vertical passing through the center of the bases?

Draw in your notebook the solid of revolution generated by rotating the following plane figure around the indicated axis. What is the name of the solid obtained? Describe its main elements (bases, height and generatrix).

A soda brand wants to design a new cylindrical can. The radius of the base must be $3,5\text{ cm}$ and the height of the can $12\text{ cm}$. Calculate the amount of metal (total surface area) needed to manufacture one of these cans. Round the result to two decimal places.

For a costume party, Chloe is making cone-shaped hats. If each hat has a slant height of $25\text{ cm}$ and the base diameter is $14\text{ cm}$, what will the height of the hat be? Use the Pythagorean theorem to find it.

Calculate the volume of a perfectly spherical scoop of ice cream that has a radius of $3\text{ cm}$. If the ice cream melts inside a cylindrical container with the same base and the same height as the sphere, what height will the liquid ice cream reach? (Assume the volume does not change when melting).

A water tank has the shape of a cylinder with a radius of $2\text{ meters}$ and a height of $5\text{ meters}$. If the tank is filled to three-quarters of its capacity, how many gallons of water does it contain? (Remember that $1\text{ ft}^3 = 1000\text{ liters}$).

A traffic cone has a height of $60\text{ cm}$ and a base radius of $15\text{ cm}$. Calculate its lateral area (the surface that would need to be painted orange). To do this, you will first need to find the slant height of the cone using the Pythagorean theorem.

A spherical orange with a diameter of $10\text{ cm}$ is cut by a plane passing at a distance of $3\text{ cm}$ from the center. When separating the two pieces, the cross-section of the cut is a circle. Calculate the radius of that circle of cut by applying the Pythagorean theorem in the shaded triangle.

We have a piece of cardboard in the shape of a circular sector like the one in the image. By joining the straight edges, we form the lateral surface of a cone. Calculate the total area of this cone knowing that the radius of the sector (which will be the slant height) is $10\text{ cm}$ and that the arc length is $18,84\text{ cm}$.

A perfume company is designing a new glass bottle. The bottle has a cylindrical base with a radius of $3 \text{ cm}$ and a height of $6 \text{ cm}$. On this base rests a frustum of a cone whose top base has a radius of $1 \text{ cm}$ and its slant height measures $5 \text{ cm}$. Knowing that the cap is a perfect hemisphere that fits exactly into the top opening with a radius of $1 \text{ cm}$, calculate the total outer surface area of the bottle (excluding the supporting base).

An industrial steel roller has the shape of a hollow cylinder. The outer diameter of the roller is $20 \text{ cm}$, the inner diameter is $16 \text{ cm}$ and its total length is $50 \text{ cm}$. If the density of the steel is $7,85 \text{ g/cm}^3$, calculate the total mass of the roller in kilograms. (Remember that to find the volume of the hollow cylinder you must subtract the volume of the inner cylinder from the outer one).

For an art exhibition, a metal piece has been built consisting of a cone that fits perfectly inside a cylinder of the same height ($12 \text{ cm}$) and the same base radius ($5 \text{ cm}$). The cone is inverted, so that its vertex touches the center of the bottom base of the cylinder. Calculate the volume of water that would be needed to fill the empty space left between the walls of the cylinder and the cone.

An ice cream cone has the shape of a cone with a height of $15 \text{ cm}$ and a radius of $3 \text{ cm}$ at the top opening. A scoop of ice cream is placed which is a perfect sphere with a radius of $3,5 \text{ cm}$. Due to its size, part of the scoop remains outside the cone and part remains inside. If the scoop rests exactly on the circular edge of the cone, what is the distance from the lowest point of the ice cream scoop to the bottom vertex of the cone? (Use the Pythagorean theorem).

A maritime buoy is formed by joining two identical cones at their bases. The total length of the buoy from tip to tip is $2 \text{ meters}$ and the diameter of the central circular cross-section where they join is $120 \text{ in}$. Calculate the total volume of the buoy in gallons and its total outer surface area.

A sphere of radius $13 \text{ cm}$ is cut by a horizontal plane passing at a distance of $5 \text{ cm}$ from the center. This cut generates a circle (cross-section). Calculate the area of that circle of cut and the area of the resulting upper spherical cap, knowing that the surface area of a spherical cap is calculated as $S = 2 \cdot \pi \cdot R \cdot h$, where $h$ is the height of the cap.

A water tank has the shape of an inverted frustum of a cone. The radius of the top base (open) is $4 \text{ meters}$ and that of the bottom base (bottom) is $2 \text{ meters}$. The total height of the tank is $6 \text{ meters}$. If the tank is filled to half its height ($3 \text{ meters}$), use similar triangles to find the radius of the water surface at that moment and calculate the volume of accumulated water.

Calculate the volume of the solid generated by rotating the following irregular polygon $360^\circ$ around the indicated vertical axis. The polygon is formed by a rectangle of $2 \times 4 \text{ cm}$ and a right triangle with a base of $3 \text{ cm}$ joined to its side.