Discover a wealth of knowledge at Westonci.ca, where experts provide answers to your most pressing questions. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.

Three natural numbers are in the ratio [tex]2: 3: 4[/tex]. If the sum of the squares of these numbers is 116, find the numbers.

Sagot :

To solve this problem, let's denote the three numbers by \(2x\), \(3x\), and \(4x\), where \(x\) is a scaling factor.

Given the sum of the squares of these numbers is 116, we can set up the following equation:
[tex]\[ (2x)^2 + (3x)^2 + (4x)^2 = 116 \][/tex]

Let's expand and simplify this equation:
[tex]\[ 4x^2 + 9x^2 + 16x^2 = 116 \][/tex]
[tex]\[ (4 + 9 + 16)x^2 = 116 \][/tex]
[tex]\[ 29x^2 = 116 \][/tex]

Next, we solve for \(x^2\) by dividing both sides of the equation by 29:
[tex]\[ x^2 = \frac{116}{29} \][/tex]
[tex]\[ x^2 = 4 \][/tex]

Taking the square root of both sides gives us two possible values for \(x\):
[tex]\[ x = 2 \quad \text{or} \quad x = -2 \][/tex]

Since we are looking for natural numbers, we consider the positive value of \(x\):
[tex]\[ x = 2 \][/tex]

Now, we use this value of \(x\) to find the three numbers:
[tex]\[ 2x = 2 \cdot 2 = 4 \][/tex]
[tex]\[ 3x = 3 \cdot 2 = 6 \][/tex]
[tex]\[ 4x = 4 \cdot 2 = 8 \][/tex]

Therefore, the three natural numbers are [tex]\(4\)[/tex], [tex]\(6\)[/tex], and [tex]\(8\)[/tex].
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.