Trapeziums Sectors and Composite Figures

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Trapeziums Sectors and Composite Figures — Free MYP5 Mathematics (Standard) Practice Questions

1QuestionEstimating 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

2QuestionArea 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

Calculate the area of one slice. [1]

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

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

Solutions

3QuestionArea 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°

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

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]

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

4QuestionArc 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

Calculate the arc length traced by the sprinkler head in each

0.5 \, \text{s}

interval. [2]

Determine the total distance travelled by the sprinkler head over the

3.0 \, \text{s}

period. [1]

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

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