Geometry of 3D Shapes (DP IB Analysis & Approaches (AA): SL): Exam Questions

A nickel earring is in the shape of a sphere with radius 4 mm.

Find the volume of the earring. Give your answer in the form $a\times {10}^k$, where $1\le a<10$ and $k$ is an integer.

The earring is melted down and reshaped to form a cylinder of height 16 mm.

Find the radius of the cylinder.

A baking container is in the shape of a cylinder, as shown in the diagram below. The diameter of the baking container is 6.7 cm and its volume, $V$, is 80 cm³.

Find the height, $h$, of the baking container.

A bowl full of cake batter has the shape of a hemisphere, as shown in the diagram below. The cake batter is poured into the baking container and fills a quarter of the container.

Find the radius, $r$, of the bowl.

Hamish is building a tree hut using cylindrical logs of length 1.1 m and radius 11.4 cm.

A wedge is cut from each log, as shown.

Find, in cm, the length of

(i) the minor arc $\mathrm{AB}$

(ii) the major arc $\mathrm{AB}$.

Find the area of the sector $\mathrm{OAB}$ that has been cut away from the end of each log.

Find the volume of each log, in cm³.

In the diagram below, $\mathrm{ABCD}$ is the square base of a right pyramid with vertex $\mathrm{V}$. The centre of the base is $\mathrm{M}$. The sides of the base are 4.2 cm and the vertical height of the pyramid, $\mathrm{VM}$, is 10.6 cm.

Find the area of triangle $\mathrm{ABV}$.

Find $\mathrm{AV}$.

Find the angle between $\left[\mathrm{AV}\right]$ and the base $\mathrm{ABCD}$.

$\mathrm{ABCDEF}$ is a pencil case in the shape of a triangular prism. The end of the pencil case is an isosceles triangle where $\mathrm{AC}=4$ cm and $\mathrm{AB}=\mathrm{BC}=6$ cm. The length of the pencil case is 10 cm. $\mathrm{M}$ is the midpoint of $\left[\mathrm{AC}\right]$. This information is shown in the diagram below.

Show that the volume of the pencil case is $80\sqrt{2}$ cm³.

Find the length of the longest pencil that could fit into the pencil case.

A coffee scoop is made from stainless steel that is 2 mm thick. It is in the shape of a hemisphere and has an outer diameter of 3.8 cm.

Show that the outer surface area of the scoop is $\frac{361}{50}\pi$ cm².

Calculate the volume of coffee that the scoop can hold, in cm³.

The density of the coffee when compacted in the scoop is 0.825 g cm⁻³. A single bag of ground coffee contains 350 g and costs $16.

Calculate the cost of the coffee in one scoop.

Two points, $\mathrm{A}\left(2,1,3\right)$ and $\mathrm{B}\left(5,2,6\right)$, are shown on a three-dimensional coordinate grid in the diagram below.

Find $\mathrm{AB}$.

Find the coordinates of the midpoint of $\left[\mathrm{AB}\right]$.

Find the angle between the line $\left(\mathrm{AB}\right)$ and the $xy$-plane.

The packaging for a firework is made from thin cardboard in the shape of a closed right cone with height 7.5 cm and base radius 5.3 cm.

Calculate the area of cardboard needed for the packaging.

Calculate the volume of the cone.

The firework company wants to use less packaging material without changing the volume of the cone.

The height of the cone is increased to 7.8 cm. Calculate the new radius of the cone. Give your answer correct to one decimal place.

Using your answer to part (c), show that the new cone needs less packaging material than the original cone.

A company makes metal doorknobs, each made of a solid cylinder of radius 27 mm and height 20 mm, topped with a solid hemisphere. The cylindrical part has a cylindrical hole in its base, of diameter 4 mm and depth 19 mm, for a screw. The axis of the hole is perpendicular to the base of the doorknob. This information is shown in the diagram below.

Show that the volume of metal needed to make one doorknob is $27626\pi$ mm³.

The cost of the metal rises to $0.00027 per mm³. To save money, the diameter of the doorknob is reduced by 5 mm. The height of the cylindrical part and the size of the hole do not change.

Calculate the amount of money saved on each doorknob. Give your answer correct to the nearest cent.

A building is to be constructed with a concrete slab foundation. In order to accommodate this foundation, a rectangular section of earth measuring 25 m by 28 m is removed to a depth of 1.3 m. The removed earth is used to create a hemispherical landscaped feature in order to reduce waste.  

Calculate the diameter of the landscaped hemisphere that can be created.

The architect has decided that a cylindrical area 2 m in height would be more appropriate than a hemisphere as a design feature.

Given that the maximum straight-line distance that is available on the site for landscaping features is 20 m, show that the cylindrical design would not be suitable for the site.

A cuboid, $\mathrm{ABCDEFGH}$, has sides of length $2x$ cm, $2\sqrt{3}x$ cm and $x$ cm.

A diagram representing this information is shown below.

Show that the length $\mathrm{BH}$ can be expressed as $\sqrt{17}x$ cm.

Find an expression, in terms of $x$, for the surface area of the cuboid.

A symmetrical candle is in the shape of a right cone with a circular base of radius $r$ cm and an initial height of $h_1$ cm. As the candle burns, its height decreases, and the candle becomes a frustum with the same base radius.

Find an expression, in terms of $h_1$, for the height of the remaining unburnt candle, $h_2$, when exactly a quarter of the initial volume has been burnt.

The height of the candle is 5 cm when a quarter of its volume has been burnt.

Show that the height of the cone that has burnt away is $\frac{5}{\sqrt[3]{4}-1}$ cm.

A number of model buildings are made in the shape of a right pyramid with a rectangular base. A single model, $\mathrm{ABCDE}$, has a base measuring 4.6 cm by 7.2 cm, and each of its sloping edges is 8.3 cm long, as shown in the diagram below.

Calculate the volume of one of the models.

The whole surface of each model is to be painted, except for the base, which is glued to the display board. An 18 ml pot of model paint covers an area of 120 cm².

Calculate the number of pots of paint that need to be bought to paint 8 of these models.

A rectangular swimming pool is to be built with length 22 m and width 14 m. The depth of the pool is 3.1 m at the deep end, rising to 1.2 m at the shallow end, as shown in the diagram below. The four vertical sides of the pool are all perpendicular to the horizontal top surface.

The pool is filled with water to a level 14 cm below the top edge of the pool.

Calculate the volume of water in the pool.

To investigate a problem with one of the walls, $\frac{2}{3}$ of the water is temporarily removed.

Find the depth of the water that is now at the deepest part of the pool.

Calculate the area of the base of the pool that is left uncovered by water.

Sphere 1 has radius $r_1$, volume $V_1$ and surface area $A_1$. Sphere 2 has radius $r_2$, volume $V_2$ and surface area $A_2$. Both spheres are made of the same material, and sphere 2 has eight times the mass of sphere 1.

The surfaces of both spheres are to be painted.

Find the ratio of the paint needed for sphere 2 to the paint needed for sphere 1.

A lunchbox has a rectangular base of length 18 cm and width 12.5 cm. The height of the box at the front, shallower end is 6.4 cm, rising to a height of 9.6 cm at the far end. There is an internal divider that is parallel to the front and the back, situated at a distance of $x$ cm from the shallower end of the box. All of the sides are perpendicular to the base.   A diagram representing this information is given below.

Calculate the total volume of the lunchbox.

Show that the volume of the front, shallower compartment can be expressed as

$10x\left(\frac{72+x}{9}\right)$ cm³

A firepit is constructed from a solid cuboid of stone, ABCDEFGH, with a length and width of 62 cm and a height of 42 cm. A depression in the shape of a hemisphere with diameter 48 cm is removed from the centre of the top face of the cuboid. M is the centre of the base of the hole. This information can be seen in the diagram below.

Calculate the distance MA.

The density of the stone is 1602 kg m⁻³.

Calculate the mass of a single firepit.

The cost of the stone is AUD 676 per m³ and labour costs AUD 38 per hour. It takes 4 hours to make one firepit. The company constructing the firepits has a budget of AUD 2000.

Calculate the number of firepits that can be constructed within budget. You need only consider the volume of stone in a completed firepit in your answer.

A spherical ball bearing of radius 4 mm is fired onto a vertical surface made of soft clay and is embedded to a depth of 2.3 mm. The shape of the ball bearing is not distorted by the impact.

A spherical cap is a portion of a sphere cut off with a plane, as can be seen in the diagram below.

The volume of a spherical cap can be calculated using the following formula: $V=\frac{1}{3}\pi h^2(3r-h)$

where $r$ is the radius of the sphere and $h$ is the height of the spherical cap.

Calculate the volume of soft clay that is displaced by the ball bearing.

Find the angle between the plane forming the cap and the radius connecting the centre of the sphere to the point where the plane intersects the surface of the sphere.

A second ball bearing is fired at the soft clay surface and is embedded to a depth of 2.8 mm. The volume of soft clay that is displaced is the same as it is in part (a).

Find the surface area of the second ball bearing.

The diagram below shows a product in the shape of a sphere with radius 3.6 cm. The product is packed in cuboidal packing crates measuring 1.7 m by 0.9 m with a depth of 22 cm. The spheres are stacked directly on top of and next to each other, using gum to fix them in position. Each layer contains the same number of spheres.

Find the number of spheres that can be packed in a single crate.

Calculate the volume of unused space between the spheres.

The company that has created the product wants to reduce costs by increasing the number of items that they can pack in one crate. They re-design the product in the shape of a cylinder keeping the radius and the volume the same.

Show that, to 3 significant figures, there is a 48.3% decrease in unused space in the packing crate if the cylindrical tube design is used instead of the spherical design, when the number of cylinders in the crate is maximised.

The diagram below shows a frustum, $\mathrm{ABCDEFGH}$, that has been made by removing a square-based pyramid of height 4 cm from a solid square-based pyramid of height 12 cm and base length 3 cm. The plane $\mathrm{EFGH}$ is parallel to the plane $\mathrm{ABCD}$.

Calculate the volume of the frustum.

Calculate the surface area of the frustum.

Find the length AG.

An ice cream cone is in the form of a right cone with a slant height of 9.1 cm and a perpendicular height of 8.7 cm. The bottom of the ice cream cone is filled with chocolate. The top of the chocolate layer is a circle of diameter 1 cm parallel to the circle forming the open top of the cone. A diagram representing this information can be seen below.

A sphere of ice cream, with radius $r$, is placed on top of the cone. It can be assumed that when the ice cream melts it will run into the cone and not down the sides until the cone is full and overflows.

Find the radius $r$ that the sphere of ice cream must have in order for it to fill the space inside the cone perfectly when melted, leaving no empty space and not overflowing.