L1 Mathematical Reasoning
Question Bank
A chocolatier makes a hollow chocolate sphere, as shown in the diagram. The sphere has an outer radius of 4 cm and an inner radius of 3.5 cm.
Calculate the volume of chocolate used to make the candy. The volume of a sphere is $V = \frac{4}{3}\pi r^3$.
A trapezoidal prism with a length of 3.1 metres has a volume of 10 m³.
What is the area of the trapezoidal face highlighted?
A box is a cuboid. Its length is $(2x + 3)$ cm, its width is $(x + 1)$ cm and its height is $(3x - 2)$ cm.
Find the total surface area of the box when $x = 4$.
A solid cone has a base radius of 7 cm and a vertical height of 24 cm.
Calculate the total surface area of the cone.
You buy an ice cream cone which has the dimensions shown in the diagram below. Disappointingly, you discover that the chocolate filling only starts at the bottom, leaving a large air gap before the ice cream scoop begins.
Assuming the ice cream is a perfect sphere that sits on top of the cone, calculate the percentage of the space inside the cone that is filled with air.
A 'hot chocolate bomb' is a hollow sphere of chocolate designed to melt in milk, as shown in the diagram. The outer radius of the bomb is 3 cm. The bomb must contain exactly $28\pi$ cm³ of chocolate.
What is the required inner radius, $r$, of the chocolate shell?
A pyramid has a rectangular base which is $12 \text{ m}$ long and $8 \text{ m}$ wide. The perpendicular height of the pyramid is $h \text{ m}$.
If the volume of the pyramid is $192 \text{ m}^3$, calculate the height, $h$, of the pyramid.
A box is a cuboid. Its length is $(x + 3)$ cm, its width is $x$ cm and its height is 4 cm.
The volume of the box is 160 cm³.
(i) Show that $x^2 + 3x - 40 = 0$.
(ii) Solve the equation to find the length and width of the box.
A pyramid has a square base of side 10 cm. Its top, T, is directly above the centre of the base, O, and its vertical height TO is 12 cm. M is the midpoint of one edge of the base.
Calculate the total surface area of the pyramid.
A rainwater tank is a cylinder with a diameter of 3.2 m and a height of 2.5 m.
(i) Calculate the volume of the tank in m³.
(ii) 1 m³ holds 1000 litres. A family uses 180 litres of water a day. How many full days will a full tank last?
A company is analysing the efficiency of its packaging for tennis balls. The diagram shows four spherical balls, each with radius $r$, packed into a cylinder.
Calculate the percentage of the space inside the can that is empty air, and show that this percentage is independent of the radius, $r$.
A metalworker has three solid metal spheres with radii of 9 cm, 12 cm, and 15 cm. She melts them all down and recasts the metal to form a single, larger solid sphere, as shown in the diagram.
What is the radius of the new, larger sphere?
An architect is calculating the volume of an attic space in a house. The cross-section of the attic is a right-angled scalene triangle, with dimensions as shown in the diagram. The attic runs the full 10.1-metre length of the building.
Calculate the volume of the attic space.
Sarah is building a garden shed with the dimensions shown in the diagram. The front of the shed is 3 m long.
(i) Calculate the depth of the shed (from front to back).
(ii) Calculate the angle the roof makes with the horizontal.
(iii) Calculate the total volume of the shed.
A rectangular box contains eight identical spheres that fit tightly inside, arranged in a single layer. The volume of each sphere is 50 cm³. Find the percentage of empty space inside the box. (Volume of a sphere $V = \frac{4}{3}\pi r^3$)
A square-base pyramid has a perpendicular height of $12 \text{ cm}$. The side length of the square base is $x \text{ cm}$.
If the volume of the pyramid is $256 \text{ cm}^3$, find the value of $x$.
A concrete garden step is a prism. Its cross-section is a trapezium with parallel sides of 40 cm and 60 cm, and a height of 30 cm. The step is 1.5 m long.
It is known that 1 m³ of concrete has a mass of 2400 kg.
Calculate the mass of the step.
An open-top box (it has no lid) has a square base of side $x$ cm and a height of 3 cm.
The box is made from 85 cm² of card: the base plus the four sides.
(i) Form a quadratic equation in $x$.
(ii) Solve it to find $x$, and then find the volume of the box.
A cone-shaped cup is partially filled with water as shown in the diagram. The water has a depth of 6 cm and a volume of 50 cm³. The total height of the cup is 9 cm.
Calculate the volume of the cup when it is full.
A construction company needs to install 16 identical hollow pipes for a drainage project, arranged as shown. Each pipe is 50 cm long and has a wall thickness of 2 cm.
The total volume of plastic material needed for all 16 pipes is exactly $16000\pi$ cm³.
By forming and solving an equation, find the inner radius, $r$, of each pipe.
An ice cream company is designing its signature product based on the "perfect" ratio of ingredients. The design, shown in the diagram below, consists of a cone, a spherical scoop of ice cream, a solid chocolate tip, and an air pocket.
Using the dimensions shown, find the simplified whole-number ratio of the volumes of Chocolate : Air : Ice Cream.
A premium chocolatier has a "golden rule" for their signature product, a hollow chocolate sphere like the one in the diagram. The rule states that the volume of the chocolate shell must be exactly equal to the volume of the hollow space inside it.
If the outer radius of the sphere is 10 cm, what must the thickness of the chocolate shell be?
A packaging designer for the tennis ball can shown in the diagram makes a surprising discovery. They claim there is a special relationship between the surface area of the can's label (the curved side) and the total surface area of the four balls packed inside.
By finding a simplified algebraic expression for both surface areas in terms of $r$, prove what this relationship is.
The Volt energy drink company is conducting a waste-space analysis for its standard packaging, shown in the diagram. A box contains 48 identical cans, each with a fixed height of 20 cm and radius, $r$.
Find the fraction of the box that is empty space, give your answer in terms of $\pi$.
For a special promotion, a chocolatier melts down their large 10 cm decorative cubes to produce the small chocolate tips for their wafer cones, with dimensions as shown in the diagram.
How many complete chocolate tips can be made from one melted-down chocolate cube?
A hollow metal sphere is melted down to create two smaller, solid spheres. The original hollow sphere has an inner radius of 22 cm and a shell that is 3 cm thick.
The larger of the two new spheres has a radius of 17 cm.
Find the radius of the smaller solid sphere.
The ancient Greek mathematician Archimedes discovered a beautiful relationship between the volumes of a cone, a sphere, and a cylinder when their dimensions are related as shown in the diagram below.
(i) Find the ratio of the volumes of Cone : Sphere : Cylinder in its simplest whole-number form.
(ii) Show algebraically that the volume of the cone plus the volume of the sphere is equal to the volume of the cylinder.
A water trough is a prism with a symmetrical trapezium as its cross-section. The parallel sides of the trapezium are $6k$ and $2k$. The non-parallel sides have an angle of 45° with the base. The length of the trough is $10k$.
(i) Merit Show that the perpendicular height of the trapezium cross-section is $2k$.
(ii) Excellence Given that the volume of the trough is 160 m³, find the value of $k$.
A shed is built in the shape of a prism. Its cross-section is an isosceles triangle. The two equal sides of the triangle are $5x$ long, and the base is $6x$ long. The length of the shed is $15x$.
(i) Merit Show that the area of the triangular cross-section is $12x^2$.
(ii) Excellence Given that the volume of the shed is 1440 m³, find the value of $x$.
A closed cylinder has a total surface area of 150π cm² and a radius of 5 cm. Find its volume.
A cylindrical glass has an inner radius of $5 \text{ cm}$ and an internal height of $22 \text{ cm}$. A total of 12 ice cubes, each with a side length of $3.5 \text{ cm}$, are placed in the empty glass.
Assuming the water level is zero before adding the ice, what will be the final height of the water in the glass once all the ice has completely melted? Assume no volume is lost in the melting process.
A hole is dug for a swimming pool. The hole is 12 m long and 5 m wide, and its side view is shown below. It is 1.2 m deep for the first 4 m, then the floor slopes down evenly to a depth of 2.4 m at the deep end.
It is known that 1 m³ of earth has a mass of 1600 kg. One truck can carry at most 18 tonnes of earth in a single load.
Calculate the number of truck loads needed to take away all the earth from the hole. Justify your answer.
A closed cylinder has a height of 7 cm and a radius of $r$ cm.
Its total surface area is $88\pi$ cm².
(i) Show that $r^2 + 7r - 44 = 0$.
(ii) Find the radius, and then calculate the volume of the cylinder.
The shape shown in the diagram is called a frustum. It is made by taking a large cone and chopping off a smaller cone from the top. The two cones are mathematically similar.
Based on the dimensions given in the diagram, calculate the height of the frustum.
The shape in the diagram is a frustum, which is created by slicing the top off a larger cone, leaving the bottom section. The smaller cone that was removed is mathematically similar to the original large cone.
Based on the dimensions provided in the diagram, calculate the height of the smaller cone that was chopped off.
The 'Frost Peak' chocolate company is retooling its production line. They want to melt down their standard 10 cm solid chocolate cubes to create exactly 500 solid chocolate tips for their wafer cones.
The machine needs to be programmed with the correct height, $h$, for the cone tip, which has a point angle of 26° as shown in the diagram.
Calculate the required height, $h$, of each chocolate tip.
A cylindrical glass has a total height of $18 \text{ cm}$, a base thickness of $1.5 \text{ cm}$, and an inner diameter of $9 \text{ cm}$. Ice cubes, each with a side length of $3 \text{ cm}$, are placed in the empty glass.
What is the maximum number of ice cubes that could fit in the glass before the resulting water overflows when the ice completely melts? Assume no volume is lost in the melting process.
An ice cream company is designing its signature product. The design, shown in the diagram, has a spherical ice cream scoop on top of a cone which contains an air pocket and a solid chocolate tip.
The company's "Balance Rule" requires that the volume of the empty air pocket inside the cone is exactly three times the volume of the chocolate tip.
Given the relationship between the chocolate tip's height and radius is $h=6r$, find the values of $h$ and $r$.
A sports company packages four spherical tennis balls, each with a radius $r$, into a cylindrical can that fits them perfectly, as shown in the diagram. The company is conducting an analysis of its packaging design.
Find the simplified, whole-number ratio for the
Volume of Air : Volume of Balls : Volume of the Can.
The 'Frost Peak' ice cream company is packaging their icecreams. In each box there are 5 icecreams with three packed upright and two inverted as shown in the diagram. The icecreams can be modelled with perfect cones, each with radius $r$. The height of each cone is $9r$.
Assuming the widest parts of the icecreams fit the dimensions of the box snuggly, calculate the fraction of the box that is empty.
An ice cream company is designing a 'Perfectly Balanced' cone. The design consists of a perfect spherical scoop of ice cream with a diameter $d$, which sits on top of a wafer cone of height $h$. The cone's opening also has a diameter of $d$, as shown in the diagram.
For the cone to be 'perfectly balanced', the volume of the ice cream scoop must be exactly the same as the volume of the cone. Find the required height, $h$, of the cone in terms of its diameter, $d$.