The sum of the squares of two consecutive odd natural numbers is 3530. The sum of the numbers is:

The correct option is 84.

Let the two consecutive odd natural numbers be represented as:

$x$
and
$x+2$

where $x$ is an odd natural number.

According to the given problem, the sum of the squares of these two consecutive odd numbers is 3530. We can write this as an equation:

$x^2+{(x+2)}^2=3530$

Expanding the second term using the algebraic identity ${(a+b)}^2=a^2+2ab+b^2$, we get:

$x^2+(x^2+4x+4)=3530$

Combining the like terms:

$2x^2+4x+4=3530$

Subtracting 3530 from both sides to form a standard quadratic equation:

$2x^2+4x-3526=0$

Dividing the entire equation by 2 to simplify it:

$x^2+2x-1763=0$

To solve this quadratic equation for $x$, we look for two numbers that multiply to -1763 and add up to 2. These two numbers are 43 and -41, since:

$43\times (-41)=-1763$
and
$43+(-41)=2$

Rewriting the middle term and factoring:

$x^2+43x-41x-1763=0$

$x(x+43)-41(x+43)=0$

$(x-41)(x+43)=0$

This gives two potential solutions for $x$:

$x=41$
or
$x=-43$

Since the problem specifies that the numbers are natural numbers (which must be positive), we reject the negative value. Thus, we have:

$x=41$

The two consecutive odd natural numbers are:

$41$
and
$41+2=43$

The problem asks for the sum of these two numbers:

$\text{Sum}=41+43=84$

Thus, the sum of the numbers is 84, which matches the option 84.