Exercises: Solids of Revolution

Imagine you are a graphic designer creating a 3D logo. To do this, you rotate a rectangle with a base of $5 \text{ cm}$ and a height of $12 \text{ cm}$ around one of its longest sides. Draw the solid of revolution obtained and identify the names of its main elements (base and height), indicating the measurement of each one.

For a costume party, you want to make a conical hat. You have measured the base radius, which is $9 \text{ cm}$, and the height of the hat, which will be $12 \text{ cm}$. Use the Pythagorean theorem to calculate the slant height ($g$) you will need to draw the net on the cardboard.

A famous brand of potato chips uses cylindrical tubs. If the base radius of the tub is $4 \text{ cm}$ and its height is $20 \text{ cm}$, calculate the area of the paper label that surrounds the side of the tub (lateral area). Take $\pi \approx 3,14$.

A basketball has a diameter of $24 \text{ cm}$. What is the total surface area of leather needed to manufacture it? Remember that the radius is half the diameter and use the formula for the surface area of a sphere $A = 4 \cdot \pi \cdot r^2$. (Use $\pi \approx 3,14$).

A soda can has a cylindrical shape. Its base radius measures $3 \text{ in}$ and its height is $12 \text{ in}$. Calculate the volume of soda it can hold if it is completely filled. Express the result in cubic inches ($cm^3$).

Identify which of the following plane figures, when rotated around the indicated axis, would result in a frustum of a cone (like an inverted coffee cup). Briefly explain why you have chosen that option.

An ice cream cone has the shape of a cone. If the base area is $25 \text{ cm}^2$ and the height of the cone is $10 \text{ cm}$, calculate its volume. Remember that the volume of a cone is a third of the volume of a cylinder with the same base and height: $V = \frac{1}{3} \cdot A_{base} \cdot h$.

In a geography class, we observe a globe. If we know that the length of the equator (great circle) is approximately $62,8 \text{ cm}$, what is the radius of that sphere? (Use $\pi \approx 3,14$).

A soda company wants to design a new cylindrical can. The radius of the base must be $3,5\text{ in}$ and the height of the can $12\text{ in}$. Calculate the total area of sheet metal needed to manufacture one can (excluding the sealing tabs) and the volume of soda it can hold in fluid ounces ($1\text{ in}^3 = 1\text{ fl oz}$).

A temporary event tent is being manufactured in the shape of a perfect cone. The height of the tent from the ground to the top vertex is $4\text{ ft}$ and the diameter of the circular base is $6\text{ ft}$. How many square feet of canvas have been needed to cover only the lateral part of the tent? (Fact: First you will need to find the slant height using the Pythagorean theorem).

A famous chocolate brand sells its chocolates in a sphere-shaped box with a diameter of $10\text{ in}$. If the box is filled to $75\%$ of its capacity with liquid chocolate, how many cubic inches of chocolate are inside? Express the result rounded to two decimal places.

Calculate the volume of a piece of wood that has been formed by joining a cylinder and a hemisphere (half a sphere). The cylinder has a height of $10\text{ cm}$ and its base radius is $4\text{ cm}$. The hemisphere rests exactly on the top base of the cylinder.

An ice cream cone has the shape of a cone. The wafer has a depth (height) of $12\text{ cm}$ and the radius of the opening is $3\text{ cm}$. If the ice cream fills the inside of the cone and a perfect scoop with the same radius as the cone protrudes from the top, calculate the total volume of ice cream.

Look at the net of a cylinder. It is known that the area of one of its circular bases is $28,26\text{ cm}^2$ and that the height of the lateral rectangle is $10\text{ cm}$. Calculate:

1. a) The radius of the base.
2. b) The lateral area of the cylinder.
3. c) The total area.

(Use $\pi \approx 3,14$).

A water tank has the shape of a cylinder with a height of $5\text{ m}$. If we want the tank to have a capacity of $100.000\text{ liters}$, what must the radius of the tank's base be? (Remember that $1\text{ m}^3 = 1.000\text{ liters}$). Round the radius to two decimal places.

A wooden sphere of radius $13\text{ cm}$ is cut by a horizontal plane. If the distance from the center of the sphere to the cut is $5\text{ cm}$, what is the radius of the circle formed in the cross-section of the cut? (Apply the Pythagorean theorem in the right triangle that is formed).

An interior design company manufactures lampshades in the shape of a frustum of a cone. The radius of the top base is $12\text{ cm}$, that of the bottom base is $20\text{ cm}$ and the height of the lampshade is $15\text{ cm}$. If the special fabric material costs $\$18,50\text{ \$/m}^2$, calculate the cost of the material needed to manufacture 25 identical lampshades (bear in mind that the lampshade has no bases, it is only the lateral surface).

A steel steamroller cylinder has the shape of a cylinder with a diameter of $1,2\text{ meters}$ and a length of $2,5\text{ meters}$. Steel has a density of $7850\text{ kg/m}^3$. If the roller is hollow and the thickness of the steel is $10\text{ cm}$ all around its contour (except for the lateral faces which are considered negligible), calculate the total mass of the roller in tons.

We want to build a grain silo consisting of a cylindrical body $4\text{ m}$ high with a radius of $3\text{ m}$, finished at the bottom with a cone of the same radius and a slant height of $5\text{ m}$ to facilitate unloading. Calculate the total storage volume of the silo and the total surface area of sheet metal needed to manufacture it (including the flat top lid of the cylinder).

A glass sphere with a radius of $15\text{ in}$ is placed into a cylindrical container with a diameter of $32\text{ in}$ containing water. If the sphere is completely submerged to the bottom, how many inches will the water level in the container rise? Give your answer rounded to two decimal places.

Calculate the surface area of an "ice cream cone" made up of a right cone $12\text{ cm}$ high with a base radius of $5\text{ cm}$, topped by a perfect hemisphere that fits exactly on the base of the cone. If we want to wrap the ice cream (cone and hemisphere) with protective paper, what is the minimum surface area of paper we will need?

A wooden sphere with a radius of $20\text{ cm}$ has been cut by a plane located $12\text{ cm}$ from the center. When the two fragments are separated, two spherical caps are obtained. Calculate the total area (curved surface plus the flat circular base of the cut) of the smaller spherical cap.

(Hint: use the Pythagorean theorem to find the radius of the cross-section circle).

A perfume bottle has the shape of a cylinder $10\text{ cm}$ high. Inside, the glass has an inverted conical shape that reaches exactly to the bottom and whose base coincides with the lid of the cylinder. If the radius of the bottle is $3\text{ cm}$, calculate what percentage of the total volume of the cylinder is occupied by the perfume when the bottle is full. Does this percentage depend on the chosen radius and height? Justify your answer.

A solid of revolution is generated by rotating a right triangle with legs $6\text{ cm}$ and $8\text{ cm}$ around its hypotenuse.

1. a) Describe the resulting solid (you can draw it).
2. b) Calculate the total surface area of the generated solid.

(Hint: The solid is formed by two cones joined at their base. You will need to calculate the height of the triangle relative to the hypotenuse).