The sum of the cubes of two numbers in the ratio 3 : 4 is 5824. The sum of the numbers is :

Options

A. `(5824)^((1)/(3))`
B. 28
C. 24
D. 14

Answer

To solve the problem, we need to find two numbers in the ratio 3:4 such that the sum of their cubes equals 5824. Let's denote the two numbers as \(3x\) and \(4x\), where \(x\) is a common multiplier.

### Step 1: Express the sum of the cubes
The sum of the cubes of the two numbers can be expressed as:
\[ (3x)^3 + (4x)^3 \]

### Step 2: Calculate the cubes
Calculating the cubes, we have:
\[ (3x)^3 = 27x^3 \quad \text{and} \quad (4x)^3 = 64x^3 \]

### Step 3: Combine the cubes
Now, we can combine these:
\[ 27x^3 + 64x^3 = 91x^3 \]

### Step 4: Set up the equation
According to the problem, the sum of the cubes is equal to 5824:
\[ 91x^3 = 5824 \]

### Step 5: Solve for \(x^3\)
To find \(x^3\), we divide both sides by 91:
\[ x^3 = \frac{5824}{91} \]
Calculating the right side:
\[ x^3 = 64 \]

### Step 6: Solve for \(x\)
Now, we take the cube root of both sides:
\[ x = \sqrt[3]{64} = 4 \]

### Step 7: Find the two numbers
Now we can find the two numbers:
\[ 3x = 3 \times 4 = 12 \quad \text{and} \quad 4x = 4 \times 4 = 16 \]

### Step 8: Calculate the sum of the numbers
Finally, we find the sum of the two numbers:
\[ 12 + 16 = 28 \]

Thus, the sum of the two numbers is **28**.

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