If the ratio of diagonals of two cubes is 3:2 then the ratio of the surface areas of the two cubes respectively is :

Options

A. `5:4`
B. `9:5`
C. `9:4`
D. can't be determined

Answer

To solve the problem, we need to find the ratio of the surface areas of two cubes given the ratio of their diagonals. Let's break it down step by step.

### Step 1: Understand the relationship between the diagonal and the side of a cube.
The formula for the diagonal (d) of a cube with side length (s) is given by:
\[ d = s \sqrt{3} \]

### Step 2: Set up the ratio of the diagonals.
Given that the ratio of the diagonals of two cubes is \( \frac{3}{2} \), we can express this as:
\[ \frac{d_1}{d_2} = \frac{3}{2} \]
Where \( d_1 \) is the diagonal of Cube 1 and \( d_2 \) is the diagonal of Cube 2.

### Step 3: Substitute the formula for the diagonals.
Using the formula for the diagonal, we can write:
\[ \frac{s_1 \sqrt{3}}{s_2 \sqrt{3}} = \frac{3}{2} \]
Here, \( s_1 \) and \( s_2 \) are the side lengths of Cube 1 and Cube 2 respectively.

### Step 4: Simplify the equation.
The \( \sqrt{3} \) cancels out:
\[ \frac{s_1}{s_2} = \frac{3}{2} \]

### Step 5: Find the ratio of the surface areas.
The surface area (A) of a cube is given by:
\[ A = 6s^2 \]
Thus, the ratio of the surface areas of the two cubes is:
\[ \frac{A_1}{A_2} = \frac{6s_1^2}{6s_2^2} = \frac{s_1^2}{s_2^2} \]

### Step 6: Substitute the ratio of the sides.
From the ratio of the sides, we have:
\[ \frac{s_1}{s_2} = \frac{3}{2} \]
Now, squaring both sides gives:
\[ \left(\frac{s_1}{s_2}\right)^2 = \left(\frac{3}{2}\right)^2 = \frac{9}{4} \]

### Step 7: Conclude the ratio of the surface areas.
Thus, the ratio of the surface areas of the two cubes is:
\[ \frac{A_1}{A_2} = \frac{9}{4} \]

### Final Answer:
The ratio of the surface areas of the two cubes is \( 9:4 \).

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