Volume and Surface Area of Pyramids Cones and Spheres

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Volume and Surface Area of Pyramids Cones and Spheres — Free MYP4 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

3QuestionSurface Area of Cones and SpheresConcept Practice

2 marks~3 minCriterion A

A party hat is designed in the shape of a cone. The base has a radius of 4 cm and a vertical height of 3 cm. The manufacturer needs to know the slant height to calculate the amount of card required.

Calculate the slant height of the cone. [2]

Solutions

4QuestionVolume of Cones Spheres and HemispheresAssessment Practice

10 marks~15 minCriterion D

An engineer is designing a spherical water tower with a conical roof. The water tower consists of a hemisphere (the lower part) and a right circular cone (the roof) placed on top of the hemisphere, sharing the same circular base. The total height of the water tower (from the base of the hemisphere to the apex of the cone) is 12 m. The total volume of the water tower must be exactly 500 m³. The engineer uses the following model: The hemisphere has radius

r

(m). The cone has base radius

r

and height

h

(m). The total height:

r + h = 12

. The total volume:

\frac{2}{3}\pi r^3 + \frac{1}{3}\pi r^2 h = 500

Using the given equations, determine the radius

r

of the hemisphere (and cone). Show all working.

The engineer's model assumes the tank walls have negligible thickness. Discuss one real-world limitation of this assumption and how it would affect the actual water capacity compared to the calculated volume.

Suppose the height measurement of 12 m has a possible error of

\pm 0.1

m. Evaluate how this error could affect the calculated radius and the resulting water capacity. In your answer, comment on the potential impact on the water supply for a community relying on this tower.

Solutions

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