Surface Area and Volume of Composite Shapes

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Surface Area and Volume of Composite Shapes — Free MYP5 Mathematics (Extended) Practice Questions

1QuestionArea of Plane Figures: triangle, rectangle, square, parallelogram, rhombus, kite, trapezium, circleAssessment Practice

6 marks~9 minCriterion D

A homeowner plans to turf a rectangular garden. The measured dimensions are

8.2

m by

5.4

m, and the tape measure has a maximum absolute error of

\pm 0.05

m in each measurement.

Write down the upper bound and lower bound of each dimension. [1]

Calculate the maximum possible area and the minimum possible area of the garden. Give your answers in m², correct to 2 decimal places. [2]

Turf is sold in whole square metres only. The homeowner buys exactly

45

m². Advise the homeowner whether this purchase guarantees the entire garden is covered, regardless of the true dimensions, justifying your advice with reference to your answer from part (b). [3]

Solutions

2QuestionArea of Plane Figures: triangle, rectangle, square, parallelogram, rhombus, kite, trapezium, circleAssessment Practice

6 marks~9 minCriterion B

The diagram shows the first three figures of a pattern made from trapeziums. In Figure

n

, each trapezium has parallel sides of length

n

cm and

(n + 4)

cm and a perpendicular height of

3

cm. Figure

n

contains exactly

n

such trapeziums arranged in a row.

Write down the number of trapeziums in Figure 4. [1]

Find a rule for the total area

A

^2

of Figure

n

, giving your answer in the form

A = an^2 + bn + c

, where

a

b

, and

c

are integers. [2]

A student claims that one of the figures in this pattern has a total area of

270

^2

. Justify whether the student's claim is correct. [3]

Solutions

3QuestionArea of Plane Figures: triangle, rectangle, square, parallelogram, rhombus, kite, trapezium, circleAssessment Practice

6 marks~9 minCriterion A

The diagram shows the graph of

f(x) = x^2 + bx + c

for

-1 \le x \le 5

, where

b

and

c

are integers. The graph passes through the point

(5, 0)

and has axis of symmetry

x = 2

Write down the coordinates of the other

x

-intercept of the graph. [1]

Find the values of

b

and

c

. [3]

The horizontal line

y = k

intersects the graph of

f

at exactly one point within the domain

-1 \le x \le 5

. Justify whether there is more than one value of

k

for which this is true. [2]

Solutions

4QuestionArea of Plane Figures: triangle, rectangle, square, parallelogram, rhombus, kite, trapezium, circleAssessment Practice

7 marks~11 minCriterion C

A rectangle has a fixed perimeter of 20 cm. Let

l

represent the length (in cm) and

w

represent the width (in cm), where

l > w > 0

.

The table below shows four possible dimensions.

$l$ (cm)	9	8	7	6
$w$ (cm)	1	2	3	4
$A$ (cm²)	?	?	?	?

Complete the table by calculating the area for each pair of dimensions. Give each area in cm². [2]

A student proposes the formula

A = l(10 - l)

to model the area of any rectangle with perimeter 20 cm and length

l

.

Show that this formula is consistent with the perimeter constraint

2l + 2w = 20

, and hence find the exact value of

l

that gives the maximum area, stating the maximum area in cm². [3]

The student claims: "The maximum area always occurs exactly halfway between the two lengths for which the area equals zero, because the quadratic

A = l(10 - l)

is symmetric about its axis of symmetry."

Justify whether this claim is mathematically correct, and assess whether the maximum area is actually attainable given the constraint

l > w > 0

. [2]

Solutions

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