The sum of the length breadth and height of a cuboid is 6√3

Let be length = l, breadth = b, height = h According to question, l + b + h = 3 8 Squaring both side, we get, ( l + b + h ) 2 = 3 8 2 ⇒ l 2 + b 2 + h 2 + 2 l b + 2 b h + 2 l h = 1 4 4 4 ..... ( i ) Diagonal of cuboid = l 2 + b 2 + h 2 ​ l 2 + b 2 + h 2 ​ = 2 2 Squaring both side, we get l 2 + b 2 + h 2 = 2 2 × 2 2 = 4 8 4 Put the value l 2 + b 2 + h 2 in equation ( i ), we get 4 8 4 + 2 ( l b + b h + l h ) = 1 4 4 4 ⇒ 2 ( l b + b h + l h ) = 1 4 4 4 − 4 8 4 ∴ Total surface area of cuboid = 9 6 0 c m 2

Let the length, breadth and height of the cuboid be l, b and h, respectively. Given: l + b + h = 5 3 cm .....(1) Length of diagonal of cuboid = l 2 + b 2 + h 2 = 3 5 cm ⇒ l 2 + b 2 + h 2 = 45 cm 2 .....(2) Now, l + b + h 2 = l 2 + b 2 + h 2 + 2 l b + 2 b h + 2 h l ⇒ 2 l b + b h + h l = 5 3 cm 2 - 45 cm 2 Using 1 and 2 ⇒ 2 l b + b h + h l = 75 - 45 = 30 cm 2 Total surface area of the cuboid = 2 l b + b h + h l = 30 cm 2 The sum of the length, breadth and height of a cuboid is 5 3 cm and length of its diagonal is 3 5 cm, then its total surface area is __30 cm 3_.

SOLUTION Let the length, breadth, height be l, b, h respectively. Now, l + b + h = 19 cm Given it's diagonal = 5 √ 5 cm Diagonal = √ l 2 + b 2 + h 2= 5 √ 5 Squaring on both sides we get 125 = l 2 + b 2 + h 2 Also, T. S. A = 2(lb+bh+hl) Now, Take l + b + h = 19 Squaring on both sides ( l + b + h ) 2 = 19² l 2 + b 2 + h 2 + 2 ( lb+ bh+hl) = 361 125 + TSA = 361 TSA = 236 c m 2 Therefore, T. S. A = 236 c m 2 (c) 236 c m 2