Part (a):

This is an application of Hooke's Law . Solving for k, we get . F is the force that stretches the spring (800 lbs) and x is the displacement of the spring from rest (4 in). Thus, .

Part (b):

Now that we have our k value, we can substitute into the equation for force:

.

If we integrate force with respect to x, we get work. The limits of integration now are from rest (x=0) to the displaced distance (x=2). The integral can be written as:

.

It takes 400 Joules of work to displace the spring 2 inches.

Part (c)

Again, we would use Hooke's Law. Since we're dealing with the same spring, we would still have the same k value. Since and , we see x (the displacement) is equal to .

Thus, a 1600 lb force can displace the spring 8 in

Hope that answers your question!!!!