Exercises: 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.

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.

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 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 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 ($cm^3$).

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.

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: $V = \frac{1}{3} \cdot A_{base} \cdot h$.

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 $\pi \approx 3,14$).

Marcos 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 Marcos 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.

An artisanal cheese factory in Castilla-La Mancha makes cheeses in the shape of a right cylinder. Each whole piece has a diameter of $20\text{ cm}$ (radius $r = 10\text{ cm}$) and a height of $8\text{ cm}$. To prepare a tasting, the master cheesemaker makes a clean, vertical flat cut that goes from top to bottom right through the exact center of the bases.\n\na) Before making any numerical calculation, estimate what plane geometric figure will appear on the cut surface and whether its area will be greater than, less than, or equal to $100\text{ cm}^2$. Justify your prediction at a glance from the given dimensions.\nb) Accurately calculate the side lengths of this flat section and its actual area. Check whether your estimate was correct and explain why a cut perpendicular to the bases that passes through the axis always produces that geometric figure.

A soft drinks company wants to design a new cylindrical can. The base radius must be $3,5\text{ cm}$ and the height of the can $12\text{ cm}$. Calculate the total surface area of sheet metal needed to manufacture one can (not including the seals) and the volume of soft drink it can hold in millilitres ($1\text{ cm}^3 = 1\text{ ml}$).

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).

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

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 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.

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.

The State Meteorological Agency (AEMET) 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 ($6\text{ cm}$).

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 steel steamroller cylinder has a diameter of $1,2\text{ metres}$ and a length of $2,5\text{ metres}$. Steel has a density of $7850\text{ kg/m}^3$. If the roller is hollow and the steel thickness is $10\text{ cm}$ all around its circumference (excluding the side faces, which are considered negligible), calculate the total mass of the roller in tonnes.

A grain silo is to be constructed, consisting of a cylindrical body $4\text{ m}$ high and with a radius of $3\text{ m}$, topped at the bottom by a cone of the same radius and a slant height of $5\text{ m}$ to facilitate discharge. Calculate the total storage volume of the silo and the total surface area of sheet metal required to manufacture it (including the flat top lid of the cylinder).

A glass sphere with a radius of $15\text{ cm}$ is submerged in a cylindrical container with a diameter of $32\text{ cm}$ containing water. If the sphere is completely submerged to the bottom, by how many centimetres will the water level in the container rise? Express the result rounded to two decimal places.

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

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).

A perfume bottle is in the shape of a cylinder $10\text{ cm}$ high. Inside, the glass forms an inverted cone that reaches exactly to the bottom and whose base coincides with the top 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? Reason your answer.

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

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

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

A precision machining workshop in Bilbao needs to manufacture a conical piece of hardened steel. The technical specification requires that the slant height measure exactly $g = 10\text{ cm}$ and that the height of the piece be exactly twice the radius of its base ($h = 2\cdot r$). To calculate the required radius $r$, two members of the technical team debate which approach to take:\n\n- Sergio proposes the trial-and-error method: testing successive decimal values for $r$ ($4,1\text{ cm}$; $4,2\text{ cm}$; $4,3\text{ cm}$...), calculating in each case $h = 2\cdot r$ and checking using the Pythagorean theorem which one gives a result closest to $g = 10\text{ cm}$.\n- Irene proposes the algebraic method: substituting the condition $h = 2\cdot r$ directly into the Pythagorean relationship of the cone ($g^2 = r^2 + h^2$), obtaining an equation with a single unknown ($10^2 = r^2 + (2\cdot r)^2$) to solve for $r$ directly.\n\na) Compare both methods with reasons: which of the two is preferable in a professional engineering environment and what limitations does the other method have?\nb) Apply the algebraic method proposed by Irene: solve the equation step by step, find the radius $r$ rounded to two decimal places, and calculate the height $h$ of the piece.

On the Galician coasts, spherical marine marker buoys with an outer radius of $R = 50\text{ cm}$ are used. When floating in the water under calm sea conditions, the horizontal plane of the sea surface cuts the sphere, forming a circle that marks the waterline. Under normal conditions, the center of the buoy is located at a vertical distance of $30\text{ cm}$ above the water level.\n\na) Model the situation using a vertical cross section passing through the center of the sphere and calculate the exact radius of the waterline circle formed by the water on the buoy.\nb) Over the months, the accumulation of algae and shellfish weighs down the buoy, causing it to sink further, so that the vertical distance from the center of the buoy to the water decreases from $30\text{ cm}$ to only $14\text{ cm}$. A maintenance technician claims: «Since the distance from the center to the water has been reduced to less than half, the radius of the waterline circle at the sea surface will have become more than twice as large». Prove algebraically whether their claim is true or false by calculating the new radius, and reason how the size of a plane cross section in a sphere changes as the plane approaches its center.