Mensuration — Free IB MYP5 Mathematics (Extended) Questions

1QuestionSolving Composite Solid ProblemsConcept Practice

2 marks~3 minCriterion A

A decorative candle is modelled as a cone of base radius

r

and height

h

, with a hemisphere of the same radius

r

sitting on top.

a

State the formula for the volume of the hemisphere and the formula for the volume of the cone. [1]

b

Show that the total volume of the candle is

V_{\text{total}} = \frac{1}{3}\pi r^2(2r + h)

[1]

Solutions

2QuestionArea 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

3QuestionArea 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

4QuestionEstimating and Approximating Irregular AreasConcept Practice

2 marks~3 minCriterion A

The outline of a lake is modelled using vertical strips of equal width

h = 20

m. The strip heights (in metres) are:

y_0 = 0,\quad y_1 = 12,\quad y_2 = 18,\quad y_3 = 22,\quad y_4 = 16,\quad y_5 = 8,\quad y_6 = 0

State the trapezoidal rule formula for estimating the area of the lake, using correct mathematical notation with

h

,

y_0

,

y_6

, and

y_1

through

y_5

clearly identified, and hence calculate the estimated area. [2]

Solutions

5QuestionUsing Compound Shapes in Area CalculationsConcept Practice

3 marks~5 minCriterion C

A compound shape consists of a rectangle of fixed length 10 cm and variable width

w

cm, with a semicircle of diameter

w

attached to one of the longer sides. The total area is given by

A = 10w + \frac{\pi w^2}{8}

The graph shows

A

(cm²) against

w

(cm) for

0 \leq w \leq 12

.

a

Use the graph to estimate the total area when

w = 6

cm. [1]

b

Calculate the exact total area when

w = 6

cm. [1]

c

A designer requires the total area to be no greater than 150 cm². Use the graph to find the maximum possible width

w

, then advise the designer on whether a width of 11 cm satisfies the constraint. [1]

Solutions

6QuestionArea of Trapeziums and RhombusesConcept Practice

4 marks~6 minCriterion A

A landscape architect is designing a trapezoidal garden bed. On a coordinate grid (1 unit = 1 m), the bed is modelled by trapezium

ABCD

with vertices

A(1, 1)

,

B(7, 1)

,

C(5, 5)

, and

D(3, 5)

, where

AB \parallel CD

.

a

Calculate the lengths of

AB

and

CD

, and the perpendicular height

h

of the trapezium. [2]

b

Using

A = \frac{1}{2}(a + b)h

, calculate the area of the garden bed. [1]

c

The architect states that a garden bed smaller than 18 m² is too small for the planned planting scheme. Advise the architect whether this garden bed is suitable for the planting scheme. [1]

Solutions

7QuestionNets and Surface Area of Cubes and CuboidsConcept Practice

2 marks~3 minCriterion A

A packaging company designs a cuboid-shaped box. The net of the box shows five labelled faces: Front, Back, Top, Bottom, and Left.

Deduce the name of the missing face and justify your answer using properties of a cuboid. [2]

Solutions

8QuestionVolume of Cones Spheres and HemispheresAssessment Practice

4 marks~6 minCriterion D

A manufacturer designs paper cone cups for a water dispenser. Each cup has a fixed height of

h = 6

cm. The graph shows the volume

V

(in cm³) plotted against radius

r

(in cm) for cones with this fixed height.

a

Interpret the shape of the graph to explain the mathematical relationship between

V

and

r

. [1]

b

Using the formula

V = \dfrac{1}{3}\pi r^2 h

, calculate the volume of a cup with radius

r = 4

cm. Give your answer in terms of

\pi

. [1]

c

Each cup must hold at least 100 cm³ of water. Advise the manufacturer whether a cup of radius

r = 4

cm is suitable for production. [2]

Solutions

9QuestionComparing Volumes of Different SolidsAssessment Practice

3 marks~5 minCriterion C

A manufacturer stores liquid in spherical tanks. The graph shows volume

V

(cm

^3

) plotted against

r^3

(cm

^3

) for three spheres, with a line of best fit through the origin.

Data points:

r^3 = 8

cm

^3

,

V = 33.5

cm

^3

r^3 = 27

cm

^3

,

V = 113.1

cm

^3

r^3 = 64

cm

^3

,

V = 268.1

cm

^3

a

Calculate the gradient of the line of best fit. Interpret what this gradient represents in the formula

V = \dfrac{4}{3}\pi r^3

. [2]

b

A storage tank has radius 5 cm. The manufacturer claims it holds at least 520 cm

^3

of liquid. Using the line of best fit, justify whether this claim is correct. [1]

Solutions

10QuestionSurface Area and Volume of PyramidsAssessment Practice

5 marks~8 minCriterion B

A cube with side length

s

can be dissected into exactly six congruent square-based pyramids, each with base side length

s

and vertical height

\dfrac{s}{2}

.

a

Show that the volume of each pyramid is

\dfrac{s^3}{6}

. [1]

b

Hence justify that the volume formula

V = \dfrac{1}{3} \times \text{base area} \times \text{height}

is consistent with the result in part (a). Show all algebraic steps. [2]

c

An architect claims this formula applies to any square-based pyramid, regardless of the ratio of height to base length. Critique this claim, identifying any limitations of the dissection argument as a general proof. [2]

Solutions

11QuestionApplying 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

12QuestionChanging Units in Density ProblemsAssessment Practice

6 marks~9 minCriterion B

A materials engineer is selecting a foam for a buoyancy device. The foam must have a density below

750 \ \text{kg/m}^3

to float with sufficient clearance.

Density data for four substances:

Density in

\text{g/cm}^3

:

0.5, \ 1.0, \ 2.7, \ 8.9

Density in

\text{kg/m}^3

:

500, \ 1000, \ 2700, \ 8900

a

Deduce the conversion factor from

\text{g/cm}^3

to

\text{kg/m}^3

. [2]

b

Construct a general formula that converts a density

d

in

\text{g/cm}^3

to density

D

in

\text{kg/m}^3

. [2]

c

The foam has a density of

0.72 \ \text{g/cm}^3

. Use your formula to find its density in

\text{kg/m}^3

, then advise the engineer whether this foam should be selected for the buoyancy device. [2]

Solutions

13QuestionMulti-Step Problems with Density and VolumeAssessment Practice

6 marks~9 minCriterion D

An aerospace engineer designs a lightweight alloy by combining Metal A (density

2.70 \text{ g/cm}^3

) and Metal B (density

8.96 \text{ g/cm}^3

). The finished component has a volume of

500 \text{ cm}^3

and a target density of

4.50 \text{ g/cm}^3

. Assume perfect mixing with no voids or impurities.

a

Calculate the mass of Metal A and the mass of Metal B required to achieve the target density. [4]

b

Identify and explain two limitations of the assumptions used in this model. [1]

c

The finished component is measured and found to have a volume of

495 \text{ cm}^3

instead of

500 \text{ cm}^3

, while the total mass remains unchanged. Calculate the actual density of the component and advise the engineer whether the component should be accepted or rejected for use in the aircraft. [1]

Solutions

14QuestionCalculating Unknowns in the Density FormulaAssessment Practice

4 marks~6 minCriterion A

A materials engineer is identifying two unknown metals using a mass–volume graph. Line X passes through

(0, 0)

and

(10 \ \text{cm}^3, \ 27 \ \text{g})

. Line Y passes through

(0, 0)

and

(10 \ \text{cm}^3, \ 89 \ \text{g})

.

a

Explain how the gradient of each line gives the density of the material. [1]

b

Calculate the density of material X and the density of material Y. [2]

c

A component requires a metal with density greater than

8.0 \ \text{g/cm}^3

. The engineer has

15 \ \text{cm}^3

of material Y available. Justify whether material Y should be selected for this component, and determine the mass of the component that could be made. [1]

Solutions

15QuestionEstimating and Approximating Irregular AreasAssessment Practice

5 marks~8 minCriterion D

A river cross-section is surveyed at seven points across its 16 m width. The distances

x

(m) from the left bank and corresponding depths

y

(m) are:

x

(m): 0, 2.5, 5.0, 8.0, 11.5, 14.0, 16.0

y

(m): 0.0, 1.8, 2.4, 3.1, 2.9, 1.5, 0.0

a

Calculate the cross-sectional area of the river using the trapezoidal rule. Show all working. [3]

b

The average water velocity at this cross-section is 1.2 m/s. Calculate the volume flow rate

Q

(m

^3

/s). [1]

c

A flood warning is issued when

Q

exceeds 42 m

^3

/s. Advise the river authority whether a flood warning should be issued, and justify your advice by identifying one limitation of the trapezoidal rule that affects the reliability of this conclusion. [1]

Solutions

16QuestionSolving Real-World Problems Involving Area and PerimeterAssessment Practice

2 marks~3 minCriterion B

A landscape architect is designing a rectangular garden with a fixed perimeter of 20 metres. The length of the garden is

x

metres.

a

Deduce an expression for the area

A

(in m²) of the garden in terms of

x

only. [1]

b

The architect claims the garden achieves its maximum possible area when it is square. Justify whether this claim is correct, and state the maximum area. [1]

Solutions

17QuestionArea of Rectangles Triangles and ParallelogramsAssessment Practice

5 marks~8 minCriterion D

A homeowner plans to tile a rectangular deck measuring 6 m by 4 m, with a right-angled triangular extension attached to one of the 4 m sides. The triangle has a base of 4 m and a perpendicular height of 2 m. The homeowner purchases exactly 28 m² of tiles, based on the total deck area alone.

The tile installer states that a 5% waste factor for cutting and breakage must be added to any area calculation before purchasing.

a

Calculate the total area of tiles that should have been purchased, including the 5% waste factor. [2]

b

Advise the homeowner whether the 28 m² purchased is sufficient for this tiling project, justifying your answer with reference to both the mathematics and the practical limitations of the area model used. [3]

Solutions

18QuestionNets 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

19QuestionSurface Area of Cylinders and PrismsAssessment Practice

5 marks~8 minCriterion D

A manufacturer drills a cylindrical hole of radius 2 cm through the centre of a solid rectangular block measuring 8 cm × 6 cm × 5 cm. The hole runs parallel to the 8 cm edge, passing completely through the block. Every exposed surface of the finished block must be coated with a protective compound.

Use

\pi = 3.14

.

a

Calculate the total surface area of the original rectangular block. [1]

b

Calculate the total surface area of the finished block after the cylindrical hole has been drilled. [2]

c

The coating costs 0.08 dollars per cm². Advise the manufacturer whether a single 25-dollar tin of coating is sufficient, justifying your answer with reference to the calculated cost and any practical risk. [2]

Solutions

20QuestionNets and Surface Area of Cubes and CuboidsAssessment Practice

6 marks~9 minCriterion C

A packaging company designs closed rectangular boxes with a fixed volume of

24 \text{ cm}^3

. Minimising surface area reduces material cost.

a

Calculate the surface area of each box using

S = 2(lw + lh + wh)

. [2]

Dimensions

(l, w, h)

in cm:

(1, 4, 6)

;

(2, 3, 4)

;

(1, 3, 8)

;

(2, 2, 6)

.

b

Justify which set of dimensions minimises surface area by identifying a geometric property shared by those dimensions. [1]

c

Let the dimensions be

x

,

y

,

z

with

xyz = 24

. Show that

S = 2\!\left(xy + \dfrac{24}{x} + \dfrac{24}{y}\right)

. Hence prove, using the AM–GM inequality, that surface area is minimised when all three dimensions are equal. Advise the company whether to use the

(2, 3, 4)

box or commission a new cubic design, justifying your recommendation with reference to the theoretical minimum. [3]

Solutions

Mensuration — Free MYP5 Mathematics (Extended) Practice Questions