We are given the following information: $a+b+c = 15$ $a^2 + b^2 + c^2 = 83$ We are asked to find the value of $ab+bc+ca$.

1. Problem Description

We are given the following information:

$a+b+c = 15$

$a^2 + b^2 + c^2 = 83$

We are asked to find the value of $ab+bc+ca$.

2. Solution Steps

We can use the formula:

$(a+b+c)^2 = a^2 + b^2 + c^2 + 2(ab+bc+ca)$

Substitute the given values into the formula:

$(15)^2 = 83 + 2(ab+bc+ca)$

$225 = 83 + 2(ab+bc+ca)$

$225 - 83 = 2(ab+bc+ca)$

$142 = 2(ab+bc+ca)$

$ab+bc+ca = \frac{142}{2}$

$ab+bc+ca = 71$

3. Final Answer

The value of $ab+bc+ca$ is 71.