The number of solid cones with integer radius and integer height each having its volume numerically equal to its total surface area is

Options

A. 0
B. 1
C. 2
D. infinite vuar

Answer

B

Explanation

Leh Height of cone = h
Radius of base = r
And slant height = l , `=l=sqrt(r^(2)+h^(2))`
Given volume = surface area.
`rArr1/3pir^(2)h=pirl+pir^(2)`
`rArrrh=3l+3r rArrl=1/3(rh-3r)`
`rArrsqrt(r^(2)+h^(2))=r/3(h-3)`
`rArrr^(2)+h^(2)=(r^(2))/9(h^(2)-6h+9)`
`h^(2)=(r^(2)h^(2))/9-(2hr^(2))/3`
`rArrh=(6r^(2))/(r^(2)-9)=6+54/(r^(2)-9)`
Since h and r must be integers, and `r^(2)-g` must be a factor of 54 `r^(2)-9` must be divisible by 3
`rArrr=3k.`
`h=6+54/(9k^(2)-9)=6+6/(k^(2)-1)`
Since `Oltk^(2)-1lt6`
`rArrk=2 ` isonly value, for which h is integer.
So, `r=2xx3=6`
`h=6+6/3=8`
l = 10
is the only possible values for r and h.