Dynamic Calculus
Home > Curve sketching and problems on maxima and minima
What is the 'chicken-run' of maximum area?
A chicken farmer wishes to make a rectangular chicken-run, using an existing brick wall as one side. She has 16 metres of wire fencing. What are the dimensions of the chicken-run that will give the maximum area for her chickens?
[You can reset the applet at any time by clicking the 'circular arrows' icon in the top right corner]
Investigate the function of the 'slider' by moving it left then right. Click the 'length of wire fencing' checkbox. Confirm that for all positions of the slider, the total length of wire fencing is 16 m.
- If the width of the chicken-run is \(x\) metres, what is the simplest expression for the length? What is the simplest expression for the area of the chicken-run?
- Click the 'area of the chicken-run' checkbox. What is the maximum area of the chicken-run?
- Click the checkbox to show the graph of width versus area. What are the coordinates of the stationary point on the curve? How do the coordinates of the point relate to the solution of this problem?
- Confirm your answer to (3) using calculus. Check your work by clicking the 'show calculation of stationary point' checkbox.
- What additional steps are necessary to confirm that the stationary point is a maximum turning point of the curve? Show the necessary working.
- What are the dimensions of the chicken-run that will give the maximum area for the farmer's chickens?
- Council regulations prohibit the farmer from using the existing brick wall as one boundary of the chicken-run. What would be the maximum area of the chicken-run she would be able to build with the same length of fencing?
- Show that the rectangle with area 32 m² and minimum perimeter is a square.
- The farmer decides to purchase additional fencing to build a chicken-run with the same area as that in question (6). Calculate the minimum length of fencing she must purchase.