Volume and Surface Area of Pyramids Cones and Spheres
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Volume and Surface Area of Pyramids Cones and Spheres — Free MYP5 Mathematics (Extended) Practice Questions

1QuestionSolving Composite Solid ProblemsConcept Practice
2 marks~3 minCriterion A
A decorative candle is modelled as a cone of base radius rr and height hh, with a hemisphere of the same radius rr sitting on top.
a
State the formula for the volume of the hemisphere and the formula for the volume of the cone. [1]
b
Show that the total volume of the candle is

Vtotal=13πr2(2r+h)V_{\text{total}} = \frac{1}{3}\pi r^2(2r + h) [1]
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Solutions

2QuestionComparing Volumes of Different SolidsAssessment Practice
3 marks~5 minCriterion C
A manufacturer stores liquid in spherical tanks. The graph shows volume VV (cm3^3) plotted against r3r^3 (cm3^3) for three spheres, with a line of best fit through the origin.

Data points:
r3=8r^3 = 8 cm3^3, V=33.5V = 33.5 cm3^3
r3=27r^3 = 27 cm3^3, V=113.1V = 113.1 cm3^3
r3=64r^3 = 64 cm3^3, V=268.1V = 268.1 cm3^3
a
Calculate the gradient of the line of best fit. Interpret what this gradient represents in the formula V=43πr3V = \dfrac{4}{3}\pi r^3. [2]
b
A storage tank has radius 5 cm. The manufacturer claims it holds at least 520 cm3^3 of liquid. Using the line of best fit, justify whether this claim is correct. [1]

Solutions

3QuestionVolume of Cones Spheres and HemispheresAssessment Practice
4 marks~6 minCriterion D
A manufacturer designs paper cone cups for a water dispenser. Each cup has a fixed height of h=6h = 6 cm. The graph shows the volume VV (in cm³) plotted against radius rr (in cm) for cones with this fixed height.
a
Interpret the shape of the graph to explain the mathematical relationship between VV and rr. [1]
b
Using the formula V=13πr2hV = \dfrac{1}{3}\pi r^2 h, calculate the volume of a cup with radius r=4r = 4 cm. Give your answer in terms of π\pi. [1]
c
Each cup must hold at least 100 cm³ of water. Advise the manufacturer whether a cup of radius r=4r = 4 cm is suitable for production. [2]
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Solutions

4QuestionSurface Area and Volume of PyramidsAssessment Practice
5 marks~8 minCriterion B
A cube with side length ss can be dissected into exactly six congruent square-based pyramids, each with base side length ss and vertical height s2\dfrac{s}{2}.
a
Show that the volume of each pyramid is s36\dfrac{s^3}{6}. [1]
b
Hence justify that the volume formula V=13×base area×heightV = \dfrac{1}{3} \times \text{base area} \times \text{height} is consistent with the result in part (a). Show all algebraic steps. [2]
c
An architect claims this formula applies to any square-based pyramid, regardless of the ratio of height to base length. Critique this claim, identifying any limitations of the dissection argument as a general proof. [2]
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Solutions

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