The sum of the cubes of three numbers is 584 and the ratio of the first to second as also of second to the third is 1 : 2. What is the third number?

Options

A. 20
B. 12
C. 8
D. None of these

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Define the Variables
Let the first number be \( x \). According to the given ratios:
- The second number will be \( 2x \) (since the ratio of the first to the second is 1:2).
- The third number will be \( 4x \) (since the ratio of the second to the third is also 1:2).

### Step 2: Write the Equation for the Sum of Cubes
The sum of the cubes of these three numbers is given as 584. Therefore, we can write the equation:
\[ x^3 + (2x)^3 + (4x)^3 = 584 \]

### Step 3: Expand the Cubes
Now, we will expand the cubes:
- \( (2x)^3 = 8x^3 \)
- \( (4x)^3 = 64x^3 \)

Substituting these values into the equation gives:
\[ x^3 + 8x^3 + 64x^3 = 584 \]

### Step 4: Combine Like Terms
Combine the terms on the left side:
\[ (1 + 8 + 64)x^3 = 584 \]
\[ 73x^3 = 584 \]

### Step 5: Solve for \( x^3 \)
Now, divide both sides by 73 to isolate \( x^3 \):
\[ x^3 = \frac{584}{73} \]

### Step 6: Calculate \( x^3 \)
Calculating \( \frac{584}{73} \):
\[ x^3 = 8 \]

### Step 7: Find \( x \)
Now, take the cube root of both sides to find \( x \):
\[ x = \sqrt[3]{8} = 2 \]

### Step 8: Find the Third Number
Now that we have \( x \), we can find the third number:
\[ \text{Third number} = 4x = 4 \times 2 = 8 \]

### Final Answer
The third number is \( 8 \).

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