For the Biot problem in a slab by expanding the sin and cos term and keeping only terms up to \(\lambda^{2}\) the following approximate relation can be obtained for the eigenvalues for small Biot number:

\[\lambda=\sqrt{B i /(1+B i / 2)}\]

Verify the result.

Find the value for a Biot number of 1 , and estimate the error in the time constant if the above equation were used.

What are the corresponding expressions for a long cylinder and a sphere?