Surface Area and Volume of 3D Solids Summary
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Surface Area and Volume of 3D Solids Summary — Free MYP4 Mathematics (Standard) Practice Questions

1QuestionNets and Surface Area of Cubes and CuboidsConcept Practice
3 marks~5 minCriterion A
A packaging company designs cardboard trays. Each tray has a fixed width of 4 cm and a fixed height of 3 cm, but its length ll (in cm) varies. The graph below shows the total surface area SS (in cm²) of a closed cuboid tray as ll varies from 0 to 10 cm.
a
Show that the surface area of the cuboid can be written as S=14l+24S = 14l + 24. [1]
b
Explain why the graph of SS against ll is a straight line. [1]
c
The company requires a tray with surface area no greater than 130 cm². Justify whether a tray of length 8 cm satisfies this requirement. [1]
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2QuestionNets and Surface Area of Cubes and CuboidsAssessment Practice
4 marks~6 minCriterion D
A cereal company is designing a cuboid-shaped box with a fixed volume of 3000 cm33000 \text{ cm}^3. Three sets of dimensions (length ×\times width ×\times height) are proposed:

Set A: 10 cm×10 cm×30 cm10 \text{ cm} \times 10 \text{ cm} \times 30 \text{ cm}
Set B: 5 cm×20 cm×30 cm5 \text{ cm} \times 20 \text{ cm} \times 30 \text{ cm}
Set C: 12.5 cm×12.5 cm×19.2 cm12.5 \text{ cm} \times 12.5 \text{ cm} \times 19.2 \text{ cm}
a
Calculate the surface area of Set A and Set B. [1]
b
Calculate the surface area of Set C. [1]
c
Advise the company which set of dimensions to choose to minimise material costs, and identify two real-world factors that the surface area model does not account for, explaining why each matters. [2]
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3QuestionNets 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 cm324 \text{ cm}^3. Minimising surface area reduces material cost.
a
Calculate the surface area of each box using S=2(lw+lh+wh)S = 2(lw + lh + wh). [2]

Dimensions (l,w,h)(l, w, h) in cm: (1,4,6)(1, 4, 6); (2,3,4)(2, 3, 4); (1,3,8)(1, 3, 8); (2,2,6)(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 xx, yy, zz with xyz=24xyz = 24. Show that S=2 ⁣(xy+24x+24y)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)(2, 3, 4) box or commission a new cubic design, justifying your recommendation with reference to the theoretical minimum. [3]
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Solutions

4QuestionNets and Surface Area of Cubes and CuboidsAssessment Practice
6 marks~9 minCriterion C
A packaging company is designing a cardboard box (a cuboid) with length l=8l = 8 cm, width w=5w = 5 cm, and height h=3h = 3 cm.
a
Construct a fully labelled net of the cuboid, showing the dimensions of all six faces. [2]
b
Show that the surface area of the cuboid is 158 cm2158 \text{ cm}^2, using the formula SA=2(lw+lh+wh)SA = 2(lw + lh + wh). [2]
c
A trainee designer claims the surface area is 150 cm2150 \text{ cm}^2 and orders cardboard accordingly. Advise the packaging manager whether this order should be accepted, identifying the most likely source of the designer's error. [2]
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