Math of Space: Surface Area and Volume

A cone and a cylinder have the same base radius $r$ and the same height $h$. What is the ratio of the volume of the cone to the volume of the cylinder?

The base radius of a right circular cylinder is $3\text{ cm}$ and its height is $7\text{ cm}$. Find its total surface area.

What is the radius of a sphere whose surface area is $154\text{ cm}^2$? (Take $\pi = \frac{22}{7}$)

A classroom is $10\text{ m}$ long, $8\text{ m}$ wide, and $4\text{ m}$ high. Find the cost of whitewashing its four walls at the rate of Rs 5 per square meter.

Find the length of the longest rod that can be placed in a room of dimensions $12\text{ m} \times 9\text{ m} \times 8\text{ m}$.

A joker's cap is in the form of a right circular cone of base radius $7\text{ cm}$ and height $24\text{ cm}$. Find the area of the sheet required to make 10 such caps.

A cylindrical tube open at both ends is made of iron sheet $2\text{ cm}$ thick. If the external diameter is $16\text{ cm}$ and length is $100\text{ cm}$, find the volume of iron used.

A copper wire of diameter $4$ mm is evenly wrapped around a cylinder whose length is $24$ cm and diameter is $20$ cm so as to cover the whole curved surface area. Find the length and mass of the wire, assuming the density of copper to be $8.88$ g/cm$^3$.

An open water tank is in the form of an inverted right circular cone of height $8$ cm and radius of its top, which is open, is $5$ cm. It is filled with water up to the brim. When lead shots, each of which is a sphere of radius $0.5$ cm, are dropped into the vessel, one-fourth of the water flows out. Find the number of lead shots dropped in the vessel.

A metallic sphere of radius $10.5$ cm is melted and then recast into small cones, each of radius $3.5$ cm and height $3$ cm. Find the total number of cones formed.

If the volume of a sphere is $38808\text{ cm}^3$, find its radius.

Find the capacity in liters of a conical vessel with radius $7\text{ cm}$ and slant height $25\text{ cm}$. (Use $1000\text{ cm}^3 = 1\text{ L}$)

A capsule of medicine is in the shape of a sphere of diameter $3.5\text{ mm}$. How much medicine (in $\text{mm}^3$) is needed to fill this capsule?

The volume of a right circular cone is $9856\text{ cm}^3$. If the diameter of the base is $28\text{ cm}$, find the height of the cone.

A cuboid has a volume of $640\text{ cm}^3$. If its length is $10\text{ cm}$ and width is $8\text{ cm}$, what is its total surface area?

A hemispherical tank is made of an iron sheet $1\text{ cm}$ thick. If the inner radius is $1\text{ m}$, then find the volume of the iron used to make the tank.

A corn cob, shaped somewhat like a cone, has the radius of its broadest end as $2.1\text{ cm}$ and length (height) as $20\text{ cm}$. If each $1\text{ cm}^2$ of the surface of the cob carries an average of four grains, find how many grains you would find on the entire cob (curved surface area).

A sculptor is carving a sphere out of a solid wooden cube of side $20 \text{ cm}$. The sculptor wants the largest possible sphere. Guesstimate the volume of wood that will be wasted as shavings. (Use $\pi \approx 3.1$).

A water tank is in the shape of a cuboid with dimensions $10 \text{ m} \times 8 \text{ m} \times 6 \text{ m}$. It is currently half-full. A spherical solid buoy of radius $2 \text{ m}$ is dropped into the tank. Guesstimate the rise in the water level. (Use $\pi \approx 3$).

A city planner is looking at a cylindrical tunnel of length $100 \text{ m}$. The inner diameter is guesstimated at $10 \text{ m}$. To prevent seepage, the inner curved surface needs a coating of special polymer. If one bucket of polymer covers $250 \text{ m}^2$, how many buckets should the planner order? (Use $\pi \approx 3.14$).