Exercises: Solids of revolution

Imagine you have a rectangular piece of cardstock with a base of $10\text{ cm}$ and a height of $15\text{ cm}$. If you rotate it quickly around one of its $15\text{ cm}$ sides, what solid of revolution is generated? Draw the resulting solid and state what its radius and height measure.

A standard soda can has a cylindrical shape. If its base radius 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 (curved surface area). Use $\pi \approx 3,14$.

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 the Pythagorean theorem to calculate the length of the slant height ($g$) of the cone.

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 the result in $\text{cm}^2$.

Calculate the volume of a cylindrical water tank that has a radius of $2\text{ meters}$ and a height of $5\text{ meters}$. If $1\text{ ft}^3$ is equivalent to $1000\text{ liters}$, how many gallons 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 protruding).

If we cut an orange in half, we obtain two hemispheres. If the radius of the orange is $4\text{ cm}$, calculate the volume of one of these halves.

Hint: Calculate the volume of the whole sphere first and then divide by two.

Look at the following net of a solid of revolution.

1. a) What solid is it?
2. b) If the radius of the circle is $R = 5\text{ cm}$ and the height of the rectangle is $H = 10\text{ cm}$, calculate the total surface area of the figure.

A photography studio uses a cone-shaped reflector to concentrate 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 needed to manufacture it (excluding the base cover).

A vitamin supplement brand 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 millimeters.

A glass 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 glass? Round the result to two decimal places.

Calculate the lateral area and the volume of a frustum of a cone (like a popcorn bucket) whose bases have radii of $10\text{ cm}$ and $6\text{ cm}$, knowing that its height is $15\text{ cm}$. Note: You will first need to calculate the slant height using the Pythagorean theorem.

A spherical gas tank has an outer diameter of $6\text{ meters}$. The engineers want to apply a coat of anti-rust paint that costs $14,50\text{ \$}$ per square foot. How much will it cost to paint the total surface of the tank? (Use $\pi \approx 3,14$).

A solid of revolution is generated by rotating a rectangle of base $4\text{ cm}$ and height $7\text{ cm}$ around one of its long sides.

1. What geometric figure is obtained?

2. Calculate its total area.

3. If it rotated around the short side, would it have the same volume? Justify your answer by performing the calculations.

In an ice cream parlor they serve a perfectly spherical scoop of ice cream of diameter $6\text{ cm}$ on a cone of height $10\text{ cm}$ and the same diameter as the scoop. If the ice cream melts and is contained exactly within the cone, will it overflow or will there be free space? Answer by comparing the volumes of the sphere and the cone.

A courier company uses cylindrical cardboard tubes to send plans. 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 manufacture the two circular caps of each tube. (Use $\pi \approx 3,14$).

A lamp designer wants to create a lampshade in the shape of a frustum of a cone. The lampshade has a height of $24 \text{ cm}$, a top diameter of $18 \text{ cm}$ and a bottom diameter of $30 \text{ cm}$. To calculate the cost of the material, they need to know the lateral surface area. However, to manufacture it from a flat piece, they must first determine the height of the original complete cone from which the frustum comes. Calculate this total height and the lateral area of the lampshade (without the bases).

An ice cream glass has the shape of an inverted cone of height $12 \text{ cm}$ and diameter $10 \text{ cm}$ at the rim. It is filled with chocolate milkshake exactly up to half its height. One student claims that the glass is "half full", but another says that there is much less than half the total volume. Calculate what percentage of the total volume of the glass the milkshake actually occupies to check who is right.

A steel water tank is to be built combining a cylinder of height $4 \text{ ft}$ and radius $3 \text{ ft}$ with a perfect hemisphere on top. If the cost of the steel is $45 \text{ \$}$ per square foot of outer surface and sealing the joint between the hemisphere and the cylinder costs $15 \text{ \$}$ per linear foot of circumference, calculate the total budget for the materials of the outer surface of the tank (including the flat bottom base).

A stone rolling pin has a cylindrical shape with a radius of $4 \text{ cm}$ and a length of $25 \text{ cm}$. It is to be stored in a cylindrical box so that the rolling pin fits exactly (tangent to the walls and bases). If it is decided to replace that rolling pin with another one that has half the radius but twice the length:

1. Will the volume of stone used change? Justify your answer with calculations.

2. Which of the two rolling pins would require a box with a larger total surface area?

Calculate the volume of the figure generated by rotating $360^\circ$ the following right trapezoid around side $AB$. Side $AD$ measures $6 \text{ cm}$ (smaller base), side $BC$ measures $10 \text{ cm}$ (larger base) and the height of the trapezoid (side $AB$) measures $8 \text{ cm}$. Identify the resulting solid and calculate its total area.

Two cities, $P$ and $Q$, are located on the same parallel at a latitude of $60^\circ \text{ N}$. City $P$ has a longitude of $15^\circ \text{ E}$ and city $Q$ of $45^\circ \text{ E}$. Knowing that the radius of the Earth is approximately $6371 \text{ km}$:

1. Calculate the radius of the parallel $60^\circ \text{ N}$ (Hint: use trigonometry or remember that in the triangle formed, the radius of the parallel is $R \cdot \cos(60^\circ)$).

2. Find the minimum distance between the two cities measured along the arc of that parallel.

A perfume bottle has the shape of a sphere of radius $R = 5 \text{ cm}$. The bottle is contained within a box in the shape of a right cylinder that has the same radius and the same height as the diameter of the sphere.

1. Calculate the difference between the volume of the cylindrical box and the volume of the sphere.

2. Show that the lateral area of the cylinder is equal to the surface area of the sphere.

A spherical lead ball of diameter $6 \text{ cm}$ has been completely submerged in a cylindrical container of radius $10 \text{ cm}$ containing water.

1. How much has the water level in the container risen after introducing the ball?

2. If after this, we melt the lead ball and manufacture a cone with it whose base has the same radius as the original sphere ($3 \text{ cm}$), what height will the cone have?