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 (Standard) Practice Questions

1QuestionVolume of Cones Spheres and HemispheresConcept Practice

2 marks~3 minCriterion C

A manufacturer designs a novelty ice-cream scoop where a single spherical scoop of radius

r

sits on top of a cone of radius

r

and height

2r

V_{\text{cone}} = \frac{1}{3}\pi r^2 h \qquad V_{\text{sphere}} = \frac{4}{3}\pi r^3

Show that

V_{\text{cone}} = \dfrac{2}{3}\pi r^3

. [1]

The manufacturer claims the sphere holds twice the ice cream of the cone. Justify whether this claim is correct. [1]

Solutions

2QuestionComparing Volumes of Different SolidsAssessment Practice

2 marks~3 minCriterion B

A cone and a cylinder share the same base radius

r

and height

h

V_{\text{cone}} = \frac{1}{3}\pi r^2 h \qquad V_{\text{cylinder}} = \pi r^2 h

Justify the relationship between

V_{\text{cone}}

and

V_{\text{cylinder}}

, using correct mathematical notation and a general statement valid for all values of

r

and

h > 0

. [2]

Solutions

3QuestionSolving Composite Solid ProblemsConcept Practice

2 marks~3 minCriterion A

A decorative candle is modelled as a cone of base radius

r

and height

h

, with a hemisphere of the same radius

r

sitting on top.

State the formula for the volume of the hemisphere and the formula for the volume of the cone. [1]

Show that the total volume of the candle is

V_{\text{total}} = \frac{1}{3}\pi r^2(2r + h)

[1]

Solutions

4QuestionSurface Area of Cones and SpheresAssessment Practice

4 marks~6 minCriterion D

A manufacturer produces party hats in the shape of a cone with a fixed base radius of 3 cm. The total surface area of a cone is given by

A = \pi r l + \pi r^2

, where

r

is the base radius and

l

is the slant height.

Show that, for this cone,

A = 3\pi l + 9\pi

. [1]

Deduce the gradient of the graph of

A

against

l

, giving your answer in terms of

\pi

. [1]

The manufacturer requires each hat to have a total surface area of no more than 120 cm². Interpret the equation from part (a) to determine the maximum slant height allowed, and advise the manufacturer on what this means for the design of the hat. [2]

Solutions

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