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MATH22981 Differential Calculus Project

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Question:

The stiffness of a given length of beam is proportional to the product of the width and the cube of the depth. Find the shape of the stiffest beam which can be cut from a cylindrical log (of the given length) with cross-sectional diameter of ????????.

A light beam starting from a point ???????? in the air takes the path of least time in order to reach a point ???????? under water. Use this principle to prove Snell’s law. Assume the speed of light is ????????/???????? in water, where ???????? is the speed of light in air and ???????? is the refractive index.

Water is flowing at the rate of ???????? into a tank in the form of a cone of altitude ? and base radius ???????? and with its vertex in the downward direction. How fast is the water level rising when the water is a certain depth ?????????

A cylindrical tank with axis horizontal has a diameter ???????? and a length ????????. It holds oil to depth ???????? (???????? < ????????) when a leak starts to drain off the oil at a constant rate ????????. How fast is the level falling?
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