Question

The lateral edge of a regular rectangular pyramid is 'a' cm long. The lateral edge makes an angle a. with the plane of the base. The value of a for which the volume of the pyramid is greatest, is ______.

Options

• `π/4`

• `sin^-1sqrt(2/3)`

• `cot^-1sqrt(2)`

• `π/3`

Answer 1

The lateral edge of a regular rectangular pyramid is 'a' cm long. The lateral edge makes an angle a. with the plane of the base. The value of a for which the volume of the pyramid is greatest, is `underlinebb(cot^-1sqrt(2))`.

Explanation:

V = `1/2 xx (2acosα)^2 xx asinα xx 1/3`

= `2/3a^3cos^2αsinα`

`(dV)/(dα)` = 0

⇒ `2/3a^3[cos^2α.cosα - sinα.2cosα.sinα]` = 0

cot²α = 2

⇒ α = `cot^-1sqrt(2)`