Paul finds an unusually shaped bowl when excavating his garden. It appears to be made out of bronze, and Paul decides to model the shape in order to work out its volume. Â
By uploading a photograph of the object onto some graphing software, Paul identifies that the cross-section of the bowl goes through the points  and . The cross-section is symmetrical about the -axis as shown in the diagram.  All of the units are in centimetres.
He models the section from  to  as a straight line.
Find the equation of the line passing through these two points.
Paul models the section of the bowl that passes through the points  and with a quadratic curve.Â
By considering the gradient of this curve when , Â explain why it may not be a good model.
Paul thinks that a quadratic with a minimum at  and passing through the point  is a better option.Â
Find the equation of the new model.
Believing this to be a better model for the bowl, Paul finds the volume of revolution about the -axis to estimate the volume of the bowl.Â
Re-arrange the answers to parts (a) and (c) to make a function of .
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