The lengths of three unequal edges of a rectangular solids block are in GP . if the volume of the block is `216 cm^(3)` and the total surface area is `252cm ^(2)` then the length of the longest edge is

Options

A. 12cm
B. 6 cm
C. 18 cm
D. 3 cm

Answer

A

Explanation

Let the length breadth and hight of rectangular solids block is `(a)/( r) ` a and ar respectively
`therefore "volume " =(a)/( r) xxaxxar =216 cm^(3)`
`implies a^(3) =216implies a^(3)=6^(3)`
`therefore a=6`
surface area ` =2((a^(2))/(r )+a^(2)r+a^(2))=252`
`implies 2a^(2)((1)/( r) _r+1)=252`
` implies 2xx36((1+r^(2)+r)/(r ))=252`
`implies (1+r^(2)+r)/(r ) =(252)/(2xx36)`
`implies 1+r^(2)+r=(126)/(36rimplies 1+r^(2)+r=(21)/(6)r`
`implies 6+6r^(2)=21rimplies 6r^(2)-15r+6=0`
`therefore r(1)/(2),2`
For ` r=(1)/(2) : "length"=(a)/(r)=(6xx2)/(1)=12`
Breadth `=a=6`
Height `=ar=6xx(1)/(2) =3`
for r=2 : length `=(a)/(r ) =(6)/(2)=3`
Breadth `=a=6`
height `=ar=6xx2=12`