Mensuration — Free IB MYP5 Mathematics (Standard) Questions

1QuestionVolume of Cones Spheres and HemispheresConcept Practice

2 marks~3 minCriterion C

A manufacturer designs a novelty ice-cream scoop where a single spherical scoop of radius

r

sits on top of a cone of radius

r

and height

2r

.

V_{\text{cone}} = \frac{1}{3}\pi r^2 h \qquad V_{\text{sphere}} = \frac{4}{3}\pi r^3

a

Show that

V_{\text{cone}} = \dfrac{2}{3}\pi r^3

. [1]

b

The manufacturer claims the sphere holds twice the ice cream of the cone. Justify whether this claim is correct. [1]

Solutions

2QuestionSolving 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

3QuestionEstimating 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

4QuestionArea 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

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

6QuestionSurface Area of Cylinders and PrismsConcept Practice

6 marks~9 minCriterion A

A manufacturer produces open-topped gift boxes by cutting and folding cardboard. Each box is a rectangular prism with a fixed base of

4\ \text{cm}

by

6\ \text{cm}

and a variable height of

h\ \text{cm}

, where

h

is a positive integer. The total surface area

S

(in

\text{cm}^2

) of the closed box is given by

S = 2(4 \times 6) + 2(4h) + 2(6h) = 48 + 20h.

a

Calculate

S

for

h = 1, 2, 3, 4, 5, 6

and record your results as a sequence. [2]

b

Deduce the surface area when

h = 10

by identifying the pattern in your sequence. Do not use the formula directly. [2]

c

The manufacturer requires a total surface area of at most

300\ \text{cm}^2

to keep material costs viable. Advise the manufacturer whether

h = 13

is a suitable production height, justifying your answer with a calculation and a reference to the constraint. [2]

Solutions

7QuestionReal-Life Applications of Volume and Surface AreaConcept Practice

2 marks~3 minCriterion C

A cylindrical water tank has radius

r = 3

m and height

h = 5

m. A maintenance company charges 12 dollars per m

^2

to apply a protective coating to the curved surface only.

a

State the formula for the curved surface area of a cylinder and calculate the curved surface area of this tank. [1]

b

The maintenance company quotes a total cost of 1131 dollars. Justify whether this quote is consistent with the given charge rate. [1]

Solutions

8QuestionComparing 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

9QuestionSurface Area of Cones and SpheresAssessment Practice

4 marks~6 minCriterion D

A manufacturer produces party hats in the shape of a cone with a fixed base radius of 3 cm. The total surface area of a cone is given by

A = \pi r l + \pi r^2

, where

r

is the base radius and

l

is the slant height.

a

Show that, for this cone,

A = 3\pi l + 9\pi

. [1]

b

Deduce the gradient of the graph of

A

against

l

, giving your answer in terms of

\pi

. [1]

c

The manufacturer requires each hat to have a total surface area of no more than 120 cm². Interpret the equation from part (a) to determine the maximum slant height allowed, and advise the manufacturer on what this means for the design of the hat. [2]

Solutions

10QuestionChanging Units in Density ProblemsAssessment Practice

4 marks~6 minCriterion C

A graph of mass (kg) against volume (m³) for a sample of aluminium produces a straight line through the origin. The gradient of the line is

2700 \ \text{kg/m}^3

.

a

State the conversion factors needed to change kg to g and m³ to cm³. [1]

b

Show that

1 \ \text{kg/m}^3 = 0.001 \ \text{g/cm}^3

, and hence calculate the density of aluminium in g/cm³. [2]

c

A student claims that because the numerical value decreased from 2700 to 2.7, the density itself has decreased. Critique this claim. [1]

Solutions

11QuestionCalculating Unknowns in the Density FormulaAssessment Practice

4 marks~6 minCriterion D

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

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

4 marks~6 minCriterion A

A cylindrical rod is made by fusing a 12 cm aluminium section to an 8 cm brass section. Both sections have radius 1.5 cm. The density of aluminium is 2.70 g/cm³ and the density of brass is 8.50 g/cm³. An engineer requires the composite rod to have an overall density greater than 4.50 g/cm³ to meet a structural specification.

a

Calculate the total volume of the composite rod. [1]

b

Calculate the total mass of the composite rod. [2]

c

Advise the engineer whether the rod meets the structural specification, justifying your answer with the calculated overall density. [1]

Solutions

14QuestionArc Length and Sector Area CalculationsAssessment Practice

5 marks~8 minCriterion C

A sprinkler arm of radius

0.8 \, \text{m}

rotates about a fixed pivot. The angular displacement of the sprinkler head is recorded below.

Time (s):

0.0,\ 0.5,\ 1.0,\ 1.5,\ 2.0,\ 2.5,\ 3.0

Angular displacement (rad):

0.0,\ 1.2,\ 2.5,\ 3.9,\ 5.4,\ 7.0,\ 8.7

a

Calculate the arc length traced by the sprinkler head in each

0.5 \, \text{s}

interval. [2]

b

Determine the total distance travelled by the sprinkler head over the

3.0 \, \text{s}

period. [1]

c

A gardener claims the sprinkler rotates at a constant angular speed of

2.9 \, \text{rad/s}

. Justify whether this claim is valid, and explain what the actual motion means for the area of ground the sprinkler covers per unit time. [2]

Solutions

15QuestionArea of Circular Sectors and SegmentsAssessment Practice

5 marks~8 minCriterion B

A circle has radius

r

and centre

O

. A sector of the circle has central angle

\theta°

.

a

State the arc length of the sector and the area of the full circle. [1]

b

A sector is dissected into many thin triangular strips, each with vertex at

O

, height approximately

r

, and base along the arc. The strips are rearranged into a shape that approximates a rectangle. Show that the area of the sector is

\dfrac{\theta}{360}\pi r^2

. [2]

c

A designer claims that a sector with

\theta = 120°

and

r = 8

cm has sufficient area to cover a surface of

165\ \text{cm}^2

. Assess whether this claim is correct, justifying your answer with a calculation. [2]

Solutions

16QuestionArea of Circular Sectors and SegmentsAssessment Practice

4 marks~6 minCriterion D

A circular pizza has a radius of 20 cm and is cut into 6 equal slices, each shaped like a sector. The crust is a 2 cm wide border along the curved edge of each slice, forming a region between the outer arc (radius 20 cm) and an inner arc (radius 18 cm).

Use

A = \dfrac{\theta}{360°} \pi r^2

.

a

Calculate the area of one slice. [1]

b

Calculate the area of the crust on one slice. [2]

c

The pizzeria claims that the crust makes up less than 20% of a slice. Justify whether this claim is correct. [1]

Solutions

17QuestionArea of Trapeziums and RhombusesAssessment Practice

5 marks~8 minCriterion B

A landscape architect is designing a trapezoidal flower bed with parallel sides of length

a

(top) and

b

(bottom), where

b > a

, and perpendicular height

h

.

a

Construct a clearly labelled diagram showing the trapezium decomposed into one rectangle and two right-angled triangles. Label the parallel sides

a

and

b

, the height

h

, and the horizontal bases of the two triangles

x

(left) and

y

(right). [1]

b

Using your diagram, justify that

x + y = b - a

, write algebraic expressions for the area of each component shape, and show that their sum simplifies to

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

. [3]

c

The architect states that a trapezoidal bed with

a = 3

m,

b = 7

m, and

h = 4

m has sufficient area to plant 20 lavender bushes, each requiring at least

0.9

m² of space. Assess whether the architect's claim is correct. [1]

Solutions

18QuestionArea 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

19QuestionNets and Surface Area of Cubes and CuboidsAssessment Practice

6 marks~9 minCriterion B

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

20QuestionSurface 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

Mensuration — Free MYP5 Mathematics (Standard) Practice Questions