The Second Perfume Bottle (Mathematics)

[Horiana]

A perfume company is designing a second new perfume bottle.

The bottle is a cylinder.
The cylinder has height 5 cm and radius 2 cm.
The lid is a sphere with radius 1 cm.

Two bands of gold paint 6 mm wide go around the bottle.

The perfume company wants answers to these questions:

Part 1: Specifications
- They need to know the total area of gold that will be painted on each bottle in order to estimate the cost of this paint.
- The bottle is to hold 50 ml of perfume. Explain why this bottle is sufficiently big to hold this amount of liquid.

Part 2: Packaging
They are going to package the bottle i a cuboid box. The dimensions of the box will be the same width and depth as the base of the bottle and the total height of the bottle.
- They would like to know the volume inside the packaging that will be wasted because it will be empty.
- They need to know the amount of cardboard needed to make the box. Allow 50% more for construction.

Part 3: Designing a new bottle
They wanted to design a new bottle. The new bottle must be sufficiently big to hold 50 ml of perfume and have the same lid.
- Design a different perfume bottle and show that it meets the perfume company's specifications. Give a sketch of your new design, including the bands of gold paint, and clearly show the dimensions of your design. (You can use any different shape such as a cylinder, a prism, sphere or a pyramid, for the bottom part of the bottle).
- Compare the use of gold paint, amount of cardboard needed and wasted packaging space of your new design with the design of the original bottle, and comment on which design would be best, and why.

[factor-fiction]

part 1- use the formulae for surface area and volume of a cylinder

[factor-fiction]

*... and a sphere..

For part two you need to be able to work out the area of rectangles and volume of a cuboid.

Look up the formulae and break the problem down in stages. It's all within the scope of GCSE Maths.

[Horiana]

So what will be the answers

[sunny-dave]

It's easy enough - as factor says, you just need to apply the formulae for volumes and areas that you will have from a GCSE course.

[Prudie]

It is actually quite straightforward although quite long, I have just enjoyed doing it when I should be working. You need the formulae for volume of a cylinder and a cuboid. Curved surface area of a cylinder and surface area of a box, add in a bit of logical thought and simple arithmetic. For your own designed bottle pick a shape that you understand the geometry of.
Totally pointless giving you the answers.

[Horiana]

Its so i know how to do it so i can learn the methods

[jim360]

The methods are as described. Get the relevant formulas, that you will have learned, and put the numbers where they belong, and do that several times.

[Peter Pedant]

Long quezzy

isnt this a variant of the gold=plated baked bean can problem we all did in Calculus ? takes me back to 1966

The idea is to have max volume for min surface area

[Prudie]

It's far simpler than that PP and doesn't involve calculus or anything harder than multiplication ( + logic)