What would be the weight of a 59.1-kg astronaut on a planet with the same density as earth and having twice earth's radius?

You could always directly use F = G M m / r², but that takes forever. 

It is best to know that, on Earth's surface, the astronaut would weigh 
F = m g = 579.2 N 
where m = 59.1 kg and g = 9.8 m/s². 

You then notice from F = G M m / r² that doubling the mass of the earth would double his weight and doubling the radius would give 1/4 his original weight. So... 
F = 1/4 * 2 * 579.2 N = 290 N