Fins are used to enhance the cooling of an object. Fins are manufactured in a wide variety of shapes and sizes, and for a mass-product, it may be beneficial to optimize the shape of the fin. Typically, the optimization seeks to find the dimensions of the fin that achieve a maximum
thermal efficiency at a minimum cost. The cost is usually expressed as the mass of the material.
Optimization of a single cylindrical fin requires finding the value of β satisfying
10β = 3 sinh(2β ).
Find the value of β > 0 that optimizes the cylindrical fin.