Mensuration — Free IB MYP4 Mathematics (Extended) Questions

1QuestionSurface Area and Volume of PyramidsConcept Practice

4 marks~6 minCriterion A

A monument consists of a square-based pyramid sitting on top of a rectangular prism. The rectangular prism has length 12 m, width 8 m, and height 5 m. The pyramid has a square base of side length 8 m and a perpendicular height of 6 m. A construction engineer states that any monument with a total volume exceeding 620 m³ requires a reinforced foundation.

a

Calculate the volume of the rectangular prism. [1]

b

Calculate the total volume of the monument. [2]

c

Advise the engineer whether a reinforced foundation is required for this monument. Justify your answer. [1]

Solutions

2QuestionChanging Units in Density ProblemsConcept Practice

2 marks~3 minCriterion B

The density of three substances is given in two unit systems.

Water:

1.00 \ \text{g/cm}^3

and

1000 \ \text{kg/m}^3

Aluminium:

2.70 \ \text{g/cm}^3

and

2700 \ \text{kg/m}^3

Lead:

11.3 \ \text{g/cm}^3

and

11300 \ \text{kg/m}^3

a

Deduce the relationship between a density value expressed in

\text{g/cm}^3

and the same density expressed in

\text{kg/m}^3

. [1]

b

A sample of iron has a density of

7.87 \ \text{g/cm}^3

. Justify whether this density is greater or less than

8000 \ \text{kg/m}^3

. [1]

Solutions

3QuestionMulti-Step Problems with Density and VolumeConcept Practice

2 marks~3 minCriterion A

A manufacturer produces solid aluminium blocks for industrial shelving. Each block is a rectangular prism with length 8 cm, width 5 cm, and an unlabelled height. The density of aluminium is

2.5 \text{ g/cm}^3

and each block has a mass of 300 g.

A storage shelf has a maximum load of 250 g per block.

a

Calculate the height of the block. [1]

b

Advise the manufacturer whether this block is suitable for the shelf. Justify your answer. [1]

Solutions

4QuestionArea of Circular Sectors and SegmentsConcept Practice

2 marks~3 minCriterion B

A circle has centre

O

and radius

r

. Sector

OAB

has central angle

\theta°

.

a

Explain why sector

OAB

represents the fraction

\dfrac{\theta}{360}

of the full circle. [1]

b

Justify why the area of sector

OAB

is given by

A = \dfrac{\theta}{360} \times \pi r^2

. [1]

Solutions

5QuestionCombining Shapes Semi-Circles Rectangles TrapeziumsConcept Practice

2 marks~3 minCriterion A

A landscape architect designs a garden bed in the shape of a rectangle with a semicircle attached to one of its shorter ends. The rectangle measures 8 cm by 4 cm. The semicircle's diameter equals the width of the rectangle.

A total of 40 cm² of specialist planting fabric is available to cover the garden bed.

Calculate the total area of the combined shape, using

\pi = 3.14

. Justify whether the available fabric is sufficient to cover the garden bed. [2]

Solutions

6QuestionArea of Circular Sectors and SegmentsConcept Practice

2 marks~3 minCriterion C

A landscape architect designs a decorative stone paving feature. One section is a circular sector with radius 8 cm and area

32\pi

cm

^2

.

a

Calculate the central angle

\theta

of the sector in degrees. [1]

b

The architect states that this sector is a semicircle, so it can be paired with an identical piece to form a complete circle with no wasted material. Justify whether the architect's statement is correct. [1]

Solutions

7QuestionArea of Rectangles Triangles and ParallelogramsConcept Practice

5 marks~8 minCriterion A

A landscape architect is designing a paved courtyard consisting of two rectangles and a right triangle.

Rectangle A: width

= (2x + 1)

cm, height

= (x - 2)

cm

Rectangle B: width

= (x + 3)

cm, height

= (x - 2)

cm

Right triangle: base

= (x + 3)

cm, height

= (x - 2)

cm

Rectangle A sits directly above Rectangle B; the right triangle is attached to the right side of Rectangle B. All dimensions are in centimetres.

a

Calculate the area of each shape when

x = 5

. [3]

b

Calculate the total paved area of the courtyard when

x = 5

. [1]

c

The architect states that a courtyard smaller than

65 \text{ cm}^2

is too small for the intended use. Advise the architect whether this courtyard is suitable, justifying your answer using your result from part (b). [1]

Solutions

8QuestionArea of Rectangles Triangles and ParallelogramsConcept Practice

4 marks~6 minCriterion B

A landscape architect is designing a parallelogram-shaped lawn with a base of 6 m and a perpendicular height of 4 m. She explains to her client that cutting a triangular section from one end of the lawn and repositioning it at the other end produces a rectangular shape with the same area.

a

Calculate the area of the rectangular shape formed. [1]

b

Deduce a general formula for the area of any parallelogram in terms of its base

b

and perpendicular height

h

. [1]

c

A rectangular patio of area 20 m² is planned alongside the lawn. The architect claims the lawn has a greater area than the patio. Justify whether this claim is correct. [2]

Solutions

9QuestionSurface Area of Cylinders and PrismsConcept Practice

3 marks~5 minCriterion A

A manufacturer produces cylindrical tins for storing loose-leaf tea. Each tin has a radius of 4 cm and a height of 10 cm. The total surface area of a cylinder is given by

2\pi r^2 + 2\pi rh

. Use

\pi = 3.14

.

a

Calculate the area of the two circular bases of the tin. [1]

b

Calculate the total surface area of the tin. [1]

c

A rectangular sheet of decorative paper measuring 30 cm by 28 cm is available to wrap the tin entirely. Justify whether this sheet is large enough to cover the total surface area of the tin. [1]

Solutions

10QuestionVolume of Cones Spheres and HemispheresAssessment Practice

10 marks~15 minCriterion D

An engineer is designing a spherical water tower with a conical roof. The water tower consists of a hemisphere (the lower part) and a right circular cone (the roof) placed on top of the hemisphere, sharing the same circular base. The total height of the water tower (from the base of the hemisphere to the apex of the cone) is 12 m. The total volume of the water tower must be exactly 500 m³. The engineer uses the following model: The hemisphere has radius

r

(m). The cone has base radius

r

and height

h

(m). The total height:

r + h = 12

. The total volume:

\frac{2}{3}\pi r^3 + \frac{1}{3}\pi r^2 h = 500

.

a

Using the given equations, determine the radius

r

of the hemisphere (and cone). Show all working.

b

The engineer's model assumes the tank walls have negligible thickness. Discuss one real-world limitation of this assumption and how it would affect the actual water capacity compared to the calculated volume.

c

Suppose the height measurement of 12 m has a possible error of

\pm 0.1

m. Evaluate how this error could affect the calculated radius and the resulting water capacity. In your answer, comment on the potential impact on the water supply for a community relying on this tower.

Solutions

11QuestionSurface Area and Volume of PyramidsAssessment Practice

3 marks~5 minCriterion C

A company manufactures decorative square pyramid gift boxes. The net of each pyramid has a square base with side length

s

cm and four triangular faces, each with slant height

l

cm. The total surface area is:

SA = s^2 + 4 \times \frac{1}{2} \times s \times l

The standard box has

s = 4

cm and

l = 6

cm. A larger box doubles the base side length to

s = 8

cm, with slant height unchanged at

l = 6

cm.

a

Calculate the total surface area of the standard box. [1]

b

Calculate the total surface area of the larger box. [1]

c

The company claims that doubling the base side length doubles the amount of card needed. Assess whether this claim is correct. [1]

Solutions

12QuestionComparing Volumes of Different SolidsAssessment Practice

2 marks~3 minCriterion B

A cone and a cylinder share the same base radius

r

and height

h

.

V_{\text{cone}} = \frac{1}{3}\pi r^2 h \qquad V_{\text{cylinder}} = \pi r^2 h

Justify the relationship between

V_{\text{cone}}

and

V_{\text{cylinder}}

, using correct mathematical notation and a general statement valid for all values of

r

and

h > 0

. [2]

Solutions

13QuestionMulti-Step Problems with Density and VolumeAssessment Practice

2 marks~3 minCriterion D

A jeweller melts 50 g of gold (density 19.3 g/cm³) and 30 g of silver (density 10.5 g/cm³) to produce an alloy ring.

a

Calculate the total volume of the alloy. [1]

b

The jeweller claims the alloy ring qualifies for hallmarking as a "high-density gold alloy," which requires a minimum density of 15.0 g/cm³. Using your result from part (a), determine the density of the alloy and advise the jeweller whether the ring meets this requirement. [1]

Solutions

14QuestionApplying Density to Real-World SituationsAssessment Practice

4 marks~6 minCriterion C

A materials scientist measures the mass and volume of samples of an unknown alloy. The graph shows a straight line through the origin and the point

(2, 8)

, where the horizontal axis represents volume (

\text{m}^3

) and the vertical axis represents mass (

\text{kg}

).

a

Explain how the gradient of this graph represents the density of the alloy. [1]

b

Calculate the density of the alloy. [1]

c

Titanium alloys used in aerospace components must have a density below

5 \ \text{kg/m}^3

to meet structural weight limits. Advise whether this alloy should be approved for use in aerospace structural components, justifying your answer with reference to the density threshold. [2]

Solutions

15QuestionArc Length and Sector Area CalculationsAssessment Practice

6 marks~9 minCriterion D

A satellite orbits Earth in a circular path with a radius of 6900 km. A sensor records data points at 12° intervals along its path. The arc length between consecutive data points is predicted using

s = r\theta

, where

\theta

is in radians.

a

Show that 12° is equal to

\dfrac{\pi}{15}

radians. [1]

b

Calculate the predicted arc length between consecutive data points. Round your answer to the nearest kilometre. [2]

The actual measured arc lengths for three consecutive intervals are:

Interval 1	actual: 1442 km
Interval 2	actual: 1448 km
Interval 3	actual: 1440 km

c

For each interval, calculate the percentage error between the actual and predicted arc lengths, using:

\text{Percentage error} = \frac{|\text{actual} - \text{predicted}|}{\text{predicted}} \times 100\%

Round each answer to one decimal place. [2]

d

A mission engineer states: "The circular orbit model is reliable enough for navigation purposes." Using your results from parts (b) and (c), advise the engineer whether the circular orbit model should be accepted for navigation use. [1]

Solutions

16QuestionArea of Rectangles Triangles and ParallelogramsAssessment Practice

5 marks~8 minCriterion C

A landscape architect is designing a paved courtyard. Two paving designs are proposed: one uses parallelogram-shaped tiles with base

a

and slant side

b

; the other uses rectangular tiles with side lengths

a

and

b

. The diagram shows both shapes, with the parallelogram's perpendicular height

h

labelled.

a

State the area formula for each tile shape. [2]

b

Explain why the two tile shapes do not generally cover the same area, even though they share the dimensions

a

and

b

. Refer to

h

,

b

, and the concept of perpendicular height in your answer. [2]

c

The architect claims the two designs are interchangeable for any tile dimensions. Justify whether this claim is correct. [1]

Solutions

17QuestionArea of Rectangles Triangles and ParallelogramsAssessment Practice

6 marks~9 minCriterion D

A rooftop garden is being designed on a parallelogram-shaped plot. The plot has a base of

8

m, a perpendicular height of

5

m, and slanted side lengths of

7

m.

a

Decompose the parallelogram into one rectangle and two congruent right-angled triangles by dropping perpendiculars from each top vertex to the base. Use the Pythagorean theorem to find the horizontal leg

x

of each right-angled triangle. [2]

b

Calculate the area of the rectangle and the combined area of the two right-angled triangles. Show that their sum equals

b \times h

. [2]

c

The garden designer claims that using a steeper slanted side of

9

m (keeping base

8

m and perpendicular height

5

m unchanged) would increase the area available for planting. Advise the designer whether this change should be made, justifying your answer with reference to what the slanted side length determines about the shape. [2]

Solutions

18QuestionNets and Surface Area of Cubes and CuboidsAssessment Practice

2 marks~3 minCriterion D

A company designs a cardboard box for shipping a laptop. The box is a cuboid with dimensions 40 cm by 30 cm by 5 cm. The cost of cardboard is 0.02 dollars per

\text{cm}^2

.

Identify one assumption made when using the surface area of the cuboid to estimate the total cost of cardboard, and explain why this assumption could make the estimate inaccurate. (Do not calculate the cost.) [2]

Solutions

19QuestionNets and Surface Area of Cubes and CuboidsAssessment Practice

5 marks~8 minCriterion B

Five nets of six squares are shown in the diagram.

Net A: cross shape (central column of 4 squares, one square attached left and one right of the second square from the top)
Net B: T-shape (row of 4 squares with one square attached above and one below the second square)
Net C:

1 \times 6

strip (all 6 squares in a single row)
Net D:

2 \times 3

rectangle
Net E: L-shape (row of 3 squares with a further row of 3 squares attached perpendicularly at one end)

a

Classify each net as valid or invalid for folding into a cube. [1]

b

For each invalid net, justify why it cannot fold into a cube, referring to overlap or face-adjacency. [2]

c

A packaging engineer claims that any connected arrangement of exactly 6 squares can fold into a cube. Critique this claim, and state the precise condition a net must satisfy to be a valid cube net. [2]

Solutions

20QuestionVolume of Cubes Cuboids and CylindersAssessment Practice

4 marks~6 minCriterion C

A manufacturer drills a cylindrical hole vertically through the centre of a solid wooden cube to reduce its mass. The cube has side length 6 cm. The cylindrical hole has radius 2 cm and passes completely through the cube from top face to bottom face.

a

Calculate the volume of the cube. [1]

b

Calculate the volume of material removed by the cylindrical hole, giving your answer in the form

a\pi

cm³. [1]

c

The manufacturer claims that more than 35% of the cube's volume has been removed. Justify whether this claim is correct. [2]

Solutions

Mensuration — Free MYP4 Mathematics (Extended) Practice Questions