Dr. J's Maths.com
Where the techniques of Maths
are explained in simple terms.

Calculus - Differentiation - Applied max/min questions.
Type 4: Advanced 3D shapes - Test Yourself 1.


 

The questions on this page focus on:
1. a box in a sphere.
2. a cone in a sphere.
3. a cylinder in a sphere.
4. A cylinder in a cone.

Common strategies: Pythagoras - also similar triangles.

Box in a sphere. 1. A prism in the shape of a rectangular box has its base length twice its width. The prism is constructed within a sphere having a radius of 10 cm such that its vertices all lie on the circumference of the sphere.

Let the width of the prism be x cm.

(i) Show that the length of the diagonal of the base of the prism is cm.

(ii) Show that the height of the prism can be written as

.

(iii) Hence find the maximum volume of the prism
(answer to nearest cubic centimetre).

Answer.Max. volume = 1232 cm3.
  2.
Cone in a sphere. 3. A cone with a radius of r and a height of h is constructed inside a sphere of radius R as shown in the diagram.

(i) Prove that the volume of the cone can be expressed as

(ii) Hence show that the maximum volume of the cone is when

.

Hint:Draw another radius R from the centre of the sphere to the lower right corner of the cone :). You now have a little right angled triangle.
  4.

The diagram above shows a cone of base radius CD = r cm and height AC = h cm inscribed in a sphere of radius of 1 m. The centre of the sphere is O and <OCD = 90°.

The distance between the centre of the sphere and the base of the cone (OC) is x cm.

(i) Show that .

(ii) Hence show that the volume V cm3 of the cone can be expressed as: .

(iii) Find the radius of the largest cone which can be inscribed in the sphere (answer to the nearest mm).

Answer.Radius = 33.3 cm.
Cylinder in a sphere. 5. A cylinder of radius r and height 2h is contained within a fixed sphere of radius R.

(i) Show the volume of the cylinder can be expressed as

(ii) Hence or otherwise, show that the cylinder has maximun volume when .
 

6.

Cylinder in a cone. 7.

A cylinder is inside a cone which has a height of 4 m. The semi-vertical angle of the cone is 30°. The height of the cylinder is h m.

(i) Show that the radius of the cone is

(ii) Show that the volume V m3 of the cylinder is given by

(iii) Hence show that the cylinder has maximum volume when its height h is one third of the height of the cone.
 

8.