Answer
(a) $490mi$
(b) $\frac{y^2}{60025}-\frac{x^2}{2475} =1 $
(c) $10.1mi$
Work Step by Step
(a) As the radio signals travel at 980 ft/ms, 2640 microseconds (ms) will correspond to a distance of
$|d(P,A)-d(P,B)|=980\times2640=2587200ft=490mi$ (1mi=5280ft)
(b) Given $d(A,B)=500mi$, we have $2c=500$ and $c=250$. Since the distance difference we get above is the same as $2a$, we have $a=245mi$, and $b^2=c^2-a^2=62500-60025=2475$ and we can write the hyperbola equation as $\frac{y^2}{60025}-\frac{x^2}{2475} =1 $
(c) Let $y=c=250$, plug it in the equation, we have $\frac{250^2}{60025}-\frac{x^2}{2475} =1$, we can find a positive x-value as $x\approx10.1mi$ which gives the distance from P to A.
