Exercise taken from Caiken, pg 23

For part (a) I get (350+2549+2*6378)/2 = 7287.5 km

for part (b) ...

Used formula for conic sections $\displaystyle \frac{p}{r} = 1 + e \cos \theta$

assumed perigee when $\displaystyle \cos \theta = 0$ and apogee when $\displaystyle cos \theta = 1$.

This gives two linear equations:

p/(360+6378) - e = 1

p/(2549+6378) + e = 1

I get p = 7679.55646345356 and e = 0.139738270028726

for part (c) I get a bit stuck.

I can't think of any formula that will help without knowing the mass of the satellite.

Edit: Looks like Keplers 3rd law is a possible candidate for (c)

$\displaystyle T^2 = \frac{4 \pi^2}{G M}a^3$ where a is the semi major axis and T is the period. That's a step closer anyway.