Answer
Distances more than $35.59$ meters.
Work Step by Step
To find the range of the distances, we have to calculate the following inequality :
$T(x)\lt 300$
$\frac{500000}{x^2+400}\lt 300$
$\frac{500000}{x^2+400}-300\lt 0$
$\frac{500000}{x^2+400}-\frac{300x^2+120000}{x^2+400}\lt 0$
$\frac{500000-300x^2-120000}{x^2+400}\lt 0$
$\frac{-300x^2+380000}{x^2+400}\lt 0$
$\frac{-300x^2+380000}{x^2+400}$ For the equation to be negative, we need either numerator or denominator to be negative. But note, not both of them.
Now we have to find zeros and then signs in the intervals between the zeros :
$-300x^2+380000=0$
$-300x^2=-380000$
$3x^2=3800$
$x^2=\frac{3800}{3}$
$x=±\sqrt{\frac{3800}{3}}$
$x^2+400=0$
$x^2=-400$
This never happens, so it is always positive.
Lets simplify the zeros :
$x=±\frac{\sqrt{11400}}{3}=±\frac{10\sqrt{114}}{3}$
$x_1\approx-35.59$
$x_2\approx35.59$
And we have the following intervals :
$(-\infty, -35.59)$; $(-35.59, 35.59)$; $(35.59, +\infty)$
And the signs are as follows : negative, positive and negative
As we were looking for the intervals of $\frac{-300x^2+380000}{x^2+400}\lt 0$, we need the intervals, the range of values where it has negative values. So we have the answer :
$x∈(-\infty, -35.59)⋃(35.59, +\infty)$
But, we are asked for the distance and it cannot be negative, So, the range of distances where the temperature of the fire is $300°$ is $35.59$ meters or more.
