Question

A cubical box has edge 10 cm and another cuboidal box is
12.5 cm long, 10 cm wide and 8 cm high.
(i) Which box has smaller total surface area?
(ii) If each edge of the cube is doubled, how many times will its
T.S.A increase?​

Answer 1

Answer:

Dimensions of Cubical Box = 10 × 10 × 10

Dimensions of Cuboidal Box = 12.5 × 10 × 8

(i) Total Surface Area of a Cube = 6a²

According to the question, a = 10 cm

⇒ TSA = 6 × 10²

⇒ TSA = 6 × 100 = 600 cm²

Total Surface Area of Cuboid = 2 ( lb + bh + hl )

⇒ TSA = 2 ( 12.5 × 10 + 10 × 8 + 8 × 12.5 )

⇒ TSA = 2 ( 125 + 80 + 100 )

⇒ TSA = 2 ( 305 ) = 610 cm²

Therefore the Cubical box has a smaller surface area.

(ii) New Dimensions of Cube = 2 × a

⇒ TSA = 6 × ( 2a )²

⇒ TSA = 6 × 4a²

⇒ TSA = 24 a²

We know that a = 10. Substituting that we get,

⇒ TSA = 24 × 10²

⇒ TSA = 24 × 100 = 2400 cm²

Thus, the TSA of the new cube has increased by 4 times the old TSA of cube.

Answer 2

Answer:

${\boxed{\bold{Answer=i)TSA\:of\:cube\:is\:smaller}}}$

${\boxed{\bold{Answer=ii)4\:times\:increased}}}$

________________________

$\bf {Given} \begin{cases}\sf{Edge\:of\:cube=10\:cm} \\\sf{Dimensions\:of\:cuboid=12.5\:cm \times 10\:cm \times 8\:cm}\end{cases}$

i)

$\tt Surface\:area\:of\:cube=6{a}^{2} \\ \rightarrow\tt TSA\:of\:cube=6({10})^{2} \:{cm}^{2} \\ \rightarrow\tt TSA\:of\:cube= 600\:{cm}^{2}$

___________________________

$\tt TSA\:of\:cuboid=2(ab+bc+ca)\:{cm}^{2} \\ \rightarrow\tt TSA\:of\:cuboid=2(12.5\times 10+10\times 8+ 8 \times 12.5)\:{cm}^{2} \\ \rightarrow\tt TSA\:of\:cuboid= (2 \times 305)\:{cm}^{2} \\ \rightarrow\tt TSA\:of\:cuboid=610\:{cm}^{2}$

$\therefore\bold{ TSA\:of\:cube\:is\:smaller }$

_________________________

ii) Edge of cube is doubled .

$\therefore\tt Edge\:of\:cube=(2\times 10)\:cm \\ \rightarrow\tt Edge\:of\:cube = 20\:cm$

$\rightarrow\tt TSA = 6{(a)}^{2} \\ \rightarrow\tt TSA = 6{(20)}^{2} \\\rightarrow\tt TSA = (6 \times 400)\:{cm}^{2} \\\rightarrow\tt TSA = 2400\:{cm}^{2}$

$\rightarrow\tt TSA\:increased=\frac{2400}{600}\:times \\\rightarrow\tt TSA\:increased= 4 \:times$

$\therefore\bold{\underline{TSA\:of\:cube\:increased\:by\:4\:times}}$