IB Mathematics Applications & Interpretation Question Type 5: Calculating the Surface Area of Specific Complex Shapes Made by Combining Two or More Shapes Exercises

Determine the radius of a sphere that has the same surface area as the composite solid formed by joining a hemisphere and a right circular cone with $r=5\text{ cm}$ and $h=4\text{ cm}$. Express your answer in exact form.

A solid is formed by casting a composite shape consisting of a hemisphere of radius $5\text{ cm}$ and a cone of base radius $5\text{ cm}$ and height $4\text{ cm}$. The solid is made of a metal with density $7.8{\text{ g/cm}}^3$. Calculate the mass of the solid. Use $\pi =3.142$ and give your answer in grams to the nearest gram.

A composite solid consists of a cone with height 4 cm on top of a hemisphere with radius 5 cm. If the outside surface of the composite solid is to be painted at a cost of $0.05 per ${\text{cm}}^2$, calculate the total cost of painting the solid. Use $\pi =3.142$ and round your answer to the nearest cent.

Calculate the curved surface area of a right circular cone with radius $5\text{ cm}$ and height $4\text{ cm}$.

Calculate the total surface area of a solid composed of a hemisphere and a right circular cone joined at their bases, where the radius of both shapes is $5\text{ cm}$ and the height of the cone is $4\text{ cm}$.

Calculate the curved surface area of a hemisphere of radius $5\text{ cm}$.

Calculate the slant height of a right circular cone with radius $5\text{ cm}$ and height $4\text{ cm}$. [2]

A second composite solid is formed by joining a hemisphere and a cone, where both the radius and the height of the cone are doubled compared to the original solid (so that for the new solid $r=10\text{ cm}$ and $h=8\text{ cm}$).

Determine the ratio of the total surface area of the new solid to that of the original solid. Express your answer in simplest form.

Calculate the volume of the composite solid formed by joining a hemisphere and a right circular cone at their bases, with radius $5\text{ cm}$ and cone height $4\text{ cm}$.

A hollow shell is created by removing a smaller hemisphere of radius $4\text{ cm}$ and a smaller right circular cone of radius $4\text{ cm}$ and height $3\text{ cm}$ from the inside of the original composite solid (hemisphere + cone) with $r=5\text{ cm}$ and $h=4\text{ cm}$. Calculate the volume of the remaining material. Give your answer in terms of $\pi$.

Find the ratio of the total curved surface area to the volume of a solid formed by joining a hemisphere and a right circular cone at their bases, given that $r=5\text{ cm}$ and $h=4\text{ cm}$. Express your answer in simplest form.

The composite solid has the same volume as a right circular cylinder of radius $5\text{ cm}$. Calculate the height of this cylinder.