| course |
course_year |
question_number |
tags |
title |
year |
Variational Principles |
IB |
78 |
IB |
2017 |
Variational Principles |
|
Paper 3, Section $I$, D |
2017 |
(a) A Pringle crisp can be defined as the surface
$$z=x y \quad \text { with } \quad x^{2}+y^{2} \leqslant 1$$
Use the method of Lagrange multipliers to find the minimum and maximum values of $z$ on the boundary of the Pringle crisp and the $(x, y)$ positions where these occur.
(b) A farmer wishes to construct a grain silo in the form of a hollow vertical cylinder of radius $r$ and height $h$ with a hollow hemispherical cap of radius $r$ on top of the cylinder. The walls of the cylinder cost $£ x$ per unit area to construct and the surface of the cap costs $£ 2 x$ per unit area to construct. Given that a total volume $V$ is desired for the silo, what values of $r$ and $h$ should be chosen to minimise the cost?