Surface Area and Volume of 3D Solids Summary

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

1QuestionReal-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.

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

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

Solutions

2QuestionSurface 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

3QuestionSurface 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}

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.

Calculate

S

for

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

and record your results as a sequence. [2]

Deduce the surface area when

h = 10

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

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

4QuestionNets 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.

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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