When a strip of skeletal muscle is stimulated by an electric current it starts to develop tension. For example, if the muscle strip is being held at a constant length, it takes more and more force to hold the muscle at a constant length, as the muscle starts to pull. This force, which develops over time once the muscle is stimulated, is called the isometric tension of the muscle; an example of how isometric tension develops in a frog skeletal muscle is shown in Fig. 25.18. The Hill model for isometric tension is where p is the muscle tension, and α, a and b are positive constants.
In fact, a and b are the same constants that we discussed in Exercise 12.22, which looked at the Hill forcevelocity curve. However, p0 in this model is slightly different from that used in Exercise 12.22.

a. Use separation of variables to show that the analytic solution of the Hill model (with p(0) = 0) is −p − (p0 +

a) ln 
p0 − p p0 
= αbt.

b. From this analytic solution, give a scientific interpretation of the constant p0.

c. From the data in Fig. 25.18 choose an appropriate value of p0, choose values of a and b from Exercise 12.22, and then find a value of α that gives (at least approximately)

the blue curve in Fig. 25.18. Plot your solution.

Warning: you can’t find p directly as a function oft, i.e., you can’t find p(t)

explicitly. So how will you do the plot?

Well, can you find t as a function of p? How will this help you?