Dynamic Calculus

Home > Curve sketching and problems on maxima and minima

A piece of wire

A length of wire 8 cm long is cut into two pieces. One piece is bent to form a circle, and the other piece to form a square. Where should the cut be made if the sum of the areas of the circle and the square is to be a minimum?
[You can reset the applet at any time by clicking the 'circular arrows' icon in the top right corner]

Investigate the function of the 'sliders' by moving them left to right.

Let \(x\) be the length of wire to be bent into a circle. Write down an expression for the radius of the circle in terms of \(x\).
Write down an expression for the side length of the square in terms of \(x\).
Click the check-box 'show dimensions of circle and square' and confirm your answers to questions (1) and (2).
Write down expressions for the area of the circle, the area of the square and the sum of the areas. Click the check-box 'sum of areas' and by changing the position of the cut, find the minimum value of the sum of the areas.
Click the check-box 'show graph of \(x\) versus area'. Vary the position of the cut by moving the slider. Confirm your answer to question (4).
Using calculus, calculate the \(x\)-value of the stationary point on the curve \(y = A(x)\). Confirm your answer by clicking the appropriate checkbox.
What additional steps are required to confirm that the stationary point is a minimum turning point? What is the minimum value of the sum of the areas?
If the entire length of wire is used to form a circle, will its area be the same, less than or greater than that of a square formed with the same length of wire? Confirm your answer using the applet.
What implications would the answer to question (8) have in the building/construction industry? How might this have influenced the architect Harry Seidler in his design of the iconic Sydney building, Australia Square? [Hint: consider the relationship between the exterior surface area and the interior floor space]