The lateral edge of a regular rectangular pyramid is `ac mlongdot` The lateral edge makes an angle `alpha` with the plane of the base. Find the value of `alpha` for which the volume of the pyramid is greatest.

Answer

`alpha = tan ^(-1)(1)/(sqrt(2))`

Explanation

h= a sin `alpha and x = a cos alpha`
`x^(2)+h^(2)=a^(2)`
`therefore V=1/3^(2)h=1/32x^(2)h`
`therefore v(alpha)=2/3a^(2)cos^(2)alpha a sin alpha`
`=2/3 a^(3)sin alpha cos^(2)alpha`
Now `v(alpha)=0` then n tan `alpha = (1)/(sqrt(2))`
`therefore V_(max)=4sqrt(3a^(3))/(27)`