Curved solids

Imagine you are a graphic designer creating a 3D logo. To do so, you rotate a rectangle with a base of $5 \text{ cm}$ and a height of $12 \text{ cm}$ around one of its longer sides (the $12 \text{ cm}$ side), as shown in the figure.

1. a) Draw the solid of revolution that is obtained.

1. b) Identify the names of its main elements (base and height) and state how long each one is.

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

Look at the three plane figures A, B and C. Each one makes a full turn around the dashed axis drawn next to it. Identify which of them produces a truncated cone (frustum). Briefly explain why you chose that option.

For a fancy-dress party, you want to make a conical hat. You have measured the radius of the base, which is $9 \text{ cm}$, and the height of the hat, which will be $12 \text{ cm}$. Use Pythagoras' theorem to calculate the length of the slant height ($g$) that you will need to draw the net on card.

Cian is designing a cone-shaped party hat for his sister's birthday. In his notebook, he writes down the following: «The radius of the circular base is $r = 5\text{ cm}$ and the height of the hat is $h = 12\text{ cm}$. Therefore, to draw the cardboard template I need the slant height to measure exactly $g = 5 + 12 = 17\text{ cm}$, since the slant height connects the edge of the base to the tip of the cone by adding both distances».\n\na) Explain with reasons what conceptual error Cian made regarding the internal geometry of the cone. What plane geometric figure actually relates $r$, $h$, and $g$?\nb) Correct his calculation by determining the real length that the slant height $g$ of the hat must have.

Met Éireann, the Irish National Meteorological Service, uses rain gauges with a conical receiver funnel in many of its stations to collect rainwater. The top circular opening of the funnel has a diameter of $24\text{ cm}$ (radius $r = 12\text{ cm}$) and its slant height measures $15\text{ cm}$.\n\na) Model the vertical cross section of the funnel by identifying the characteristic right triangle and calculate the total depth (height $h$) of the cone.\nb) During a rain event, the collected water fills the funnel until it reaches a height of $4.5\text{ cm}$ from the bottom vertex (that is, exactly half of its total height). Reason geometrically what shape the free surface of the water forms and explain why its radius must be exactly half the radius of the top opening ($6\text{ cm}$).

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 $A = 4 \times \pi \times r^2$. (Use $\pi \approx 3.14$).

In a geography class, we look at 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 famous crisp brand 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 surrounding the side of the tub (curved surface area). Take $\pi \approx 3.14$.

A temporary tent for an event is being made in the shape of a perfect cone. The height of the tent from the ground to the top vertex is $4\text{ m}$ and the diameter of the circular base is $6\text{ m}$. How many square metres of canvas were needed to cover only the lateral part of the tent? (Note: You must first find the slant height using Pythagoras' theorem).

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 surface area of the cylinder.
3. c) The total surface area.

(Use $\pi \approx 3.14$).

A fizzy drink can is cylindrical in shape. The radius of its base is $3 \text{ cm}$ and its height is $12 \text{ cm}$. Calculate the volume of drink it can contain if it is filled completely. Express the result in cubic centimetres ($cm^3$).

An ice cream cone is shaped like a cone. If the area of the base 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 one-third of the volume of a cylinder with the same base and height: $V = \frac{1}{3} \times A_{base} \times h$.

A water tank is in the shape of a cylinder with a height of $5\text{ m}$. If we want the tank to have a capacity of $100,000\text{ litres}$, what radius must the base of the tank have? (Remember that $1\text{ m}^3 = 1,000\text{ litres}$). Round the radius to two decimal places.

Calculate the volume of a wooden piece formed by joining a cylinder and a hemisphere. 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 is shaped like a cone. The wafer cone 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 also sticks out as a perfect scoop on top with the same radius as the cone, calculate the total volume of ice cream.

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 make 25 identical lampshades (note that the lampshade has no top or bottom bases, only the lateral surface).

A wooden sphere with a radius of $13\text{ cm}$ is cut by a horizontal plane. If the distance from the centre 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 Pythagoras' theorem in the right-angled triangle that is formed).

In an Art class, solids of revolution are modeled with clay. When cutting a right cone with a taut wire using a vertical flat cut that passes through its top vertex and divides the circular base into two symmetrical halves, the exposed cut surface is an isosceles triangle with an area of $120\text{ cm}^2$ whose base measures $16\text{ cm}$.\n\na) Reconstruct the dimensions of the original cone: deduce with reasons the measures of the radius $r$ of its base and its height $h$, explaining which element of the cone corresponds to each side and each dimension of the observed triangle.\nb) Calculate the exact measure of the slant height $g$ of that cone.

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

(Hint: use Pythagoras' theorem to find the radius of the circular cross-section).