Dynamic Calculus

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The cylinder of maximum volume

The sum of the radius and the height of a cylinder is 5 cm. What is the maximum possible volume of the cylinder?
[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 'sum of radius and height' checkbox. Confirm that for all positions of the slider, the sum of the radius and height of the cylinder is 5 cm.

1. Write down an expression for the height of the cylinder in terms of the radius \(r\). Write down an expression for the volume of the cylinder in terms of \(r\).
2. Click the 'volume of cylinder' checkbox. What is the maximum volume of the cylinder?
3. Click the checkbox to show the graph of radius versus volume. 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?
4. Confirm your answer to (3) using calculus. Check your answer by clicking the 'show calculation of stationary point' checkbox.
5. What additional steps are needed to confirm that the stationary point is a maximum turning point of the curve? Show the necessary working.
6. What are the dimensions of the cylinder that will give the maximum volume?

State of New South Wales, Department of Education and Training 2008, Created with GeoGebra