Three numbers are in the ratio 4 : 3 : 1. If the sum of their squares is 1664, then what are the numbers?

Options

A. `32, 24, 8`
B. `32, 16, 24`
C. `4, 16, 32`
D. `28, 7, 4`

Answer

To solve the problem, we need to find three numbers that are in the ratio 4:3:1 and whose squares sum up to 1664. Let's denote these numbers as \(4x\), \(3x\), and \(x\).

### Step-by-Step Solution:

1. **Define the Numbers:**
Let the three numbers be:
\[ a = 4x, \quad b = 3x, \quad c = x \]

2. **Set Up the Equation for the Sum of Squares:**
According to the problem, the sum of their squares is given by:
\[ a^2 + b^2 + c^2 = 1664 \]
Substituting the values of \(a\), \(b\), and \(c\):
\[ (4x)^2 + (3x)^2 + (x)^2 = 1664 \]

3. **Calculate the Squares:**
Calculate each square:
\[ 16x^2 + 9x^2 + x^2 = 1664 \]

4. **Combine Like Terms:**
Combine the terms on the left side:
\[ 16x^2 + 9x^2 + 1x^2 = 26x^2 \]
Thus, we have:
\[ 26x^2 = 1664 \]

5. **Solve for \(x^2\):**
Divide both sides by 26:
\[ x^2 = \frac{1664}{26} \]
Simplifying the right side:
\[ x^2 = 64 \]

6. **Find \(x\):**
Taking the square root of both sides:
\[ x = 8 \]

7. **Calculate the Numbers:**
Now substitute \(x\) back into the expressions for \(a\), \(b\), and \(c\):
\[ a = 4x = 4 \times 8 = 32 \]
\[ b = 3x = 3 \times 8 = 24 \]
\[ c = x = 8 \]

8. **Final Answer:**
The three numbers are:
\[ 32, 24, \text{ and } 8 \]