Surface Area and Volume of 3D Solids Summary

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Surface Area and Volume of 3D Solids Summary — Free MYP5 Mathematics (Extended) Practice Questions

1QuestionNets 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

2QuestionNets 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)

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

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

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

3QuestionSurface 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

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

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

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

4QuestionNets 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.

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)

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

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

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