Answer
The "dead spot" is about 215 thousand miles from the center of the earth
Work Step by Step
1. Calculate $x$, if $F = 0$.
$$0 = \frac{-K}{x^2} + \frac{0.012K}{(239-x)^2}$$ $$\frac K {x^2} = \frac{0.012 K}{(239-x)^2}$$
- Multiply both dividend and divisor of the left side fraction by 0.012, we get:
$$\frac {0.012K} {0.012x^2} = \frac{0.012 K}{(239-x)^2}$$
Therefore:
$$0.012x^2 = (239 - x)^2$$
- Since $x$ measures distance, it can't be negative:
$$\sqrt{0.012x^2} = \sqrt{(239 - x)^2}$$ $$0.1095x = 239 - x$$ $$0.1095x + x = 239$$ $$1.1095x = 239$$ $$x = \frac{239}{1.1095} = 215$$
