Question

Bruno has a habit of collecting odd bits of “stuff”. While cleaning out his cupboard, he finds a 30cm piece of string from a present he received. He plays with the string and makes different shapes with it. He thinks that the quadrilateral with the biggest area he can make, is a square. Britney disagrees. Who is right and why?

Hints for this question include:

• Determine the maximum area of a rectangle with a given perimeter

• Consider how the area of a quadrilateral changes as its shape changes

• Interpret a relationship from a graph

Step 1. Formulate. Translate the problem into a mathematically purposeful representation.

- What data is needed? What sample size will be required? What variables will be measured?

- Identify and document any assumptions/observations you will make.

- What statistical concepts and techniques are needed to mathematise and solve this problem?

Step 2: Solve - Select and then apply the statistical (mathematical) technique that you will use for the task.

- Your approach should clearly present the relationships between variables, when they exist, or verify that they do not exist.

- You may use algebraic, graphical and/or computational methods, but the use of technology is required

Answer 1

The length of the string is given as 30 cm.

Thus the perimeter of a rectangle that can be made from the string is 30cm.

Thus l=(15-b)

Then the area of the rectangle A = (15-b)b = 15b - b2

Since the second derivative is negative , area is maximum.

Thus the quadrilateral with maximum area for given perimeter is a square and thus Bruno is right.