PART 1

A large cartoon is to be wrapped up like a birthday present using string. This investigation is about efficient wrapping using the minimum length of string for different size cartoon.

A parcel is to be tied with a length of string, assume the parcel is a cuboid with a square cross section and a fixed volume of 8000cm^3. The string goes all the way around the parcel including underneath it.

a) Find an expression for the length, Lcm of the string in terms of x only. Hint: write an expression for the volume first in terms of x and y.

b) Find the values of x and y necessary to minimize the length of the string required. What is the minimum length of the string required? How did you know that this was the minimum?

c) Sketch a graph of L against x over a suitable domain. Make sure that x is on the horizontal axis. You are allowed to use the Casio Classpad Calculator (CAS) to sketch the graph but you have to explain what steps you used on your calculator.

PART 2

The same parcel is now tied in the way shown.

In the diagram on the picture I uploaded before.

a) Find an expression for the length, ‘L’cm of the string required in terms of x only.

b) Find the dimensions of the box necessary to minimize the length of the string required.

c) Sketch the graph of L against x.

Do both methods use the same amount of string? If not then under what circumstances would both methods use the same amount of string?

PART 3

Consider the situation where the side lengths of the cuboid are ‘mx’ cm, ‘nx’ cm and ‘y’ cm; where m and n are positive integers. Assume the volume is still 8000cm^3.

a) Find an expression for the length of the string required in terms of x, m and n

b) Find, in terms of m and n, the values of ‘x’ and ‘y’ required for minimum length of string.

c) Find in terms of ‘m’ and ‘n’, the minimum length of string.

d) Now choose three different sets of values for m and n. explain why you chose these values. With each set of values draw a sketch of L against x over suitable domain. It is advisable to use same scale on each graph to make comparison earlier. With each graph, find the point at which the length of the string is the minimum. What do you notice? Can you explain this?

e) By choosing 5 different values of volume and keeping m and n constant, draw sketches of L against x, look at the points which give the minimum values of the length of string, what do you notice? Why is this?

Please refer to the picture on the part 3 I attached before