Two cubes have their volumes in the ratio 1 : 27. What is the ratio of their surface areas?
Answers
Answer:
The ratio of their surface areas of two cubes is 1: 9.
Step-by-step explanation:
SOLUTION :
Given:
Volumes of two cubes are in the ratio , V1 : V2 = 1: 27.
Let ‘a1’ & ‘a2’ be the side of first cube and second cube.
Volume of a first cube,V1 = a1³
Volume of a second cube,V2 = a2³
Ratio of Volumes of two cubes = V1 : V2 = a1³ : a2³
1 : 27 = a1³ : a2³
a1: a2 = ∛1 : ∛27
a1: a2 = 1 : 3
a1/a2 = ⅓
Surface area of a first cube ,S1 = 6a1²
Surface area of a second cube, S2 = 6a2²
Ratio of Surface area of two cubes = S1 : S2 = 6a1² : 6a2²
S1 / S2 = 6a1² : 6a2²
S1 / S2 = a1² / a2²
S1 / S2 = (a1/a2)²
S1 / S2 = (⅓)²
S1 / S2 = 1/9
S1 : S2 = 1 : 9
Hence, the ratio of their surface areas of two cubes is 1: 9.
Let their edges be a and b.
Then,
a³/b³ = 1/27
⇒ (a/b)³ = (1/3)³
⇒ a/b = 1/3 -------------------- (1)
Therefore, Ratio of the surface area,
⇒ 6a²/6b²
⇒ a²/b²
⇒ (a/b)²
⇒(1/3)² [From (1)]
⇒1/9
⇒i.e. 1 : 9
Ratio of the surface area = 1 : 9