The volume of two cubes are in the ratio 1 : 64. If the volume of the larger cube is 504 `cm^3` more than that of the smaller cube, then
(i) What is the volume of the smaller cube?
(ii) What is the length of each side of the larger cube?

Options

A. `{:(i,ii),(8"cm"^3, 8 "cm"):}`
B. `{:(i,ii),(8"cm"^3, 7 "cm"):}`
C. `{:(i,ii),(7"cm"^3, 8 "cm"):}`
D. `{:(i,ii),(7"cm"^3, 9 "cm"):}`

Answer

To solve the problem step by step, we will follow the information given in the question and the video transcript.

### Step 1: Define the volumes of the cubes
Let the volume of the smaller cube be \( V_s \) and the volume of the larger cube be \( V_l \). According to the problem, the ratio of their volumes is given as:
\[ V_s : V_l = 1 : 64 \]
This means we can express the volumes as:
\[ V_s = x \quad \text{and} \quad V_l = 64x \]

### Step 2: Set up the equation based on the difference in volumes
The problem states that the volume of the larger cube is 504 cm³ more than that of the smaller cube. This can be expressed as:
\[ V_l = V_s + 504 \]
Substituting the expressions for \( V_s \) and \( V_l \):
\[ 64x = x + 504 \]

### Step 3: Solve for \( x \)
Now, we will rearrange the equation to isolate \( x \):
\[ 64x - x = 504 \]
\[ 63x = 504 \]
Now, divide both sides by 63:
\[ x = \frac{504}{63} \]
Calculating this gives:
\[ x = 8 \]

### Step 4: Find the volume of the smaller cube
Now that we have \( x \), we can find the volume of the smaller cube:
\[ V_s = x = 8 \, \text{cm}^3 \]

### Step 5: Find the volume of the larger cube
Next, we calculate the volume of the larger cube using the value of \( x \):
\[ V_l = 64x = 64 \times 8 = 512 \, \text{cm}^3 \]

### Step 6: Find the length of each side of the larger cube
The volume of a cube is given by the formula:
\[ V = \text{side}^3 \]
So, for the larger cube:
\[ 512 = \text{side}^3 \]
To find the length of each side, we take the cube root:
\[ \text{side} = \sqrt[3]{512} \]
Calculating this gives:
\[ \text{side} = 8 \, \text{cm} \]

### Final Answers
(i) The volume of the smaller cube is \( 8 \, \text{cm}^3 \).
(ii) The length of each side of the larger cube is \( 8 \, \text{cm} \).