The length of three unequal edges of a rectangular solid block are in G.P. The volume of the block is 216 cm^{3} and the total surface area is 516 cm^{2} . The length of the longest edge is ?
+128475 Call the edges x , rx and r^2 x
So
x ( rx ) ( r^2 x) = 216 ⇒ x^3 r^3 = 216 ⇒ xr = 6 ⇒ r = 6/x
And
2 ( x * xr + x * xr^2 + xr * xr^2) = 516
x^2r + x^2r^2 + x^2r^3 = 258
x^2 (6/x) + x^2 ( 6/x)^2 + x^2 ( 6/x)^3 = 258
6x + 36 + 216/x = 258 divide through by 6
x + 6 + 36/x = 43 mutiply through by x
x^2 +6x + 36 = 43x
x^2 - 37x + 36 = 0
(x - 1) ( x - 36) = 0
x = 1 or x = 36
Take the first value and r = 6 /1 = 6
The longest edge is 1^2 * 6^2 = 36
+128475 Call the edges x , rx and r^2 x
So
x ( rx ) ( r^2 x) = 216 ⇒ x^3 r^3 = 216 ⇒ xr = 6 ⇒ r = 6/x
And
2 ( x * xr + x * xr^2 + xr * xr^2) = 516
x^2r + x^2r^2 + x^2r^3 = 258
x^2 (6/x) + x^2 ( 6/x)^2 + x^2 ( 6/x)^3 = 258
6x + 36 + 216/x = 258 divide through by 6
x + 6 + 36/x = 43 mutiply through by x
x^2 +6x + 36 = 43x
x^2 - 37x + 36 = 0
(x - 1) ( x - 36) = 0
x = 1 or x = 36
Take the first value and r = 6 /1 = 6
The longest edge is 1^2 * 6^2 = 36
