If the volume of a cube is 729 cu cm, what is the length of its diagonal?

Total surface area of a cube = 6a²
? 6 = 6a²
? a² = 1
? a = 1
Now, volume of the cube = a³ = 1³ = 1 cu unit

Given, diagonal = 2?3
? a?3 = 2?3 [a = edge of the cube]
? a = 2
? Required surface area = 6a²
= 6 x 2² = 24 sq cm

Let the height of the cylinder be H and its radius = r
Then, (?r² H) + (1/3) x (?r²h) = 3 x (1/3)?r²h
? ?r²H = (2/3)?r²h
? H = 2h/3.

Area of 4 walls of the room = [2 (l + b ) x h ]m²
Area of 4 walls of new room = [2 (3l + 3b) x 3h]m²
= 9 x [2(l + b) x h ]m²
? Cost of painting the 4 walls of the new room = Rs. (9 x 350)
= Rs. 3150

? 4/3?r³ = 4/3? x [(3/2)³ - {(3/4)³ + 1³ } ]
? r³ = 125/64 = (5/4)³
? r = 5/4
? Diameter = (5/4) x 2 cm = 2.5 cm.

Given that, l = 10 m , b = 5 m, h = 3 m
lb = 10 x 5 = 50, bh = 5 x 3 = 15,
lh = 10 x 3 = 30
? Surface area of a cuboid
= 2(lb + bh + lh)
= 2(50 + 15 + 30)
= 2 x 95 = 190 sq m

According to the question,
6a² = 726 [a = edge of the cube]
? a² = 726/6 = 121
? a = ?121 = 11 cm
? Required volume = a³ = 11³ = 1331 cm³

Volume = 1989
? lbh = 1989
? 17 x 13 x h = 1989
? h = 1989/(17 x 17) = 9 cm

Volume (new cube) = (1³ + 6³ + 8³ ) = 729 cm³
a³ = 729
? a = ? 729
? Surface area of the new cube = 6a²
= 6 x 9² = 486 sq cm
Surface area/2 = 486/2 = 243 sq cm

Length of largest pencil that can be kept in a box
= Diagonal of box = ?l² + b² + h²
where, l = 8 cm, b = 6 cm , h = 2 cm
= ?64 + 36 + 4
= ?104 = 2?26