Question

2.
Calculate the length of face diagonal of the cube whose side is 12√3 cm long.
What is the length of the space diagonal of that cube?​

Answer 1

Answer:

Then length of its diagonal is d=√{a²+a²+a²}=a√3. But, given diagonal is 6√3. Comparing both the results, we find a=6. Now, total surface area of a cube whose side length is a, T= 6a².

Answer 2

Answer:

Space diagonal = 18√2 cm

Step-by-step explanation:

PART 1

SOLVING FOR LENGTH OF THE SIDE OF THE CUBE

Let face diagonal = c = 12√3 cm

Using pythagorean theorem → c^2 = a^2 + b^2

Solving for c:

c = √a^2 + b^2

12√3 = √a^2 + a^2 → SINCE IT'S A CUBE SO THEY SAME SIDE LENGTH, SO I SUBSTITUTED "b" with "a"

(12√3)^2 = (√2a^2)^2 → TO SOLVE FOR a WE SHOULD RAISE BOTH SIDES BY THE POWER OF 2 TO ELIMINATE THE RADICAL SIGN ON "a"

144*3 = 2a^2

432 = 2a^2 → DIVIDE BOTH SIDES BY 2

432/2 = 2a^2/2

216 = a^2

216 = a^2 → EXTRACT THE SQUARE ROOT OF BOTH SIDE TO GET "a"

√216 = √a^2

Therefore, the length of the side of the CUBE is:

a = 6√6 cm

PART 2

SOLVING FOR THE SPACE DIAGONAL OF THE CUBE.

USING PYTHAGOREAN THEOREM WHERE THE SPACE DIAGONAL IS THE HYPOTHENUSE AND THE LEGS ARE THE FACE DIAGONAL'S LENGTH AND CUBE'S SIDE LENGTH.

SPACE DIAGONAL'S LENGTH = √(FACE DIAGONAL LENGTH)^2 + (CUBE'S SIDE LENGTH)^2

SPACE DIAGONAL'S LENGTH = √(12√3)^2 + (6√6)^2

SPACE DIAGONAL'S LENGTH = √(432) + (216)

SPACE DIAGONAL'S LENGTH = 18√2 cm. Q.E.D