If the volume of two cubes are in the ratio 27 : 64, then the ratio of their total surface area is :

Options

A. a)`27:64`
B. b)`3:4`
C. c)`9:16`
D. d)`3:8`

Answer

To solve the problem, we need to find the ratio of the total surface area of two cubes given the ratio of their volumes.

### Step 1: Understand the relationship between volume and side length of a cube
The volume \( V \) of a cube with side length \( a \) is given by the formula:
\[ V = a^3 \]

### Step 2: Let the side lengths of the two cubes be \( a_1 \) and \( a_2 \)
Given that the volumes of the two cubes are in the ratio 27:64, we can express this as:
\[ \frac{V_1}{V_2} = \frac{27}{64} \]
This means:
\[ \frac{a_1^3}{a_2^3} = \frac{27}{64} \]

### Step 3: Take the cube root of both sides
To find the ratio of the side lengths, we take the cube root of both sides:
\[ \frac{a_1}{a_2} = \frac{\sqrt[3]{27}}{\sqrt[3]{64}} = \frac{3}{4} \]

### Step 4: Calculate the total surface area of the cubes
The total surface area \( S \) of a cube with side length \( a \) is given by the formula:
\[ S = 6a^2 \]
Thus, the total surface areas of the two cubes are:
\[ S_1 = 6a_1^2 \quad \text{and} \quad S_2 = 6a_2^2 \]

### Step 5: Find the ratio of the total surface areas
Now we can find the ratio of the total surface areas:
\[ \frac{S_1}{S_2} = \frac{6a_1^2}{6a_2^2} = \frac{a_1^2}{a_2^2} \]
Using the ratio of the side lengths we found earlier:
\[ \frac{S_1}{S_2} = \left(\frac{a_1}{a_2}\right)^2 = \left(\frac{3}{4}\right)^2 = \frac{9}{16} \]

### Final Answer
Therefore, the ratio of their total surface areas is:
\[ \frac{9}{16} \]

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