One thing to note is that the weight measured was given in pounds instead of Newtons. We could convert the weights, but it turns out that we don't need to.

If your weight measured on Earth is

and the weight on the craft is

we know that on Earth the acceleration is . Using Newton's second law,

and

we can set  equal since the mass never changes only the weight force, and solve for ,

This makes sense that you would weigh more if you were experiencing a larger acceleration. This is interesting because you could not tell if you were standing stationary in a gravitational field or accelerating in a non-inertial reference frame. This is what is known as the Equivalence Principle.

Using Newton's second law, we can determine the answer. 

In our problem,  and .  is the acceleration due to gravity. Therefore,

The reading on a scale is really the normal force . On a flat surface, we see the person has a reading of , or . On an inclined surface, one can show that the normal force has a new magnitude of: 

This will lead the scale to read read:

Plug in values:

Solve for :

Weight force is given by:

We only need the equations for weight and normal force to solve this problem. First the equation for weight (force of gravity):

Now the normal force, which is a function of weight:

If you're unsure why we used cosine, think about the situation practically. As the hill gets flatter (angle decreased), the normal force grows, and cosine is the function that gets larger as the angle gets smaller.

From the problem statement we know that the ratio of weight to normal force is 3:

We will start this problem by using the following kinematics equation:

Note that each variable is oriented in the direction of the slope of the hill. We will find the distance traveled along the hill, and then convert that to a height using the slope of the hill.

We already know the value of each variable, so we can find the distance traveled:

Now we can use the slope of the hill and the sine function to determine the height dropped:

We know these values, so let's plug them in:

By constructing a force diagram, it can be seen that:

Right when the box starts to move, these will be equal

Solving for

Plugging in values:

Concerning the astronaut:

The force of gravity will be essentially constant and downward. Thus, in order for acceleration of the astronaut to increase, the normal force must increase.

In this problem the  force acts in the downward direction. The box's weight also acts in the downward direction. We can calculate weight of the box using the equation .  

Normal force acts in the opposition to the weight of the box. We can calculate normal force by adding up the downward forces to find the counteracting force on the box.  

Forces downward:

Forces upward: 

Forces downward are equal to the forces upward. Therefore,