Suppose a student performed an experiment to determine the concentration of a substrate as a function of time (with units of s). Denote the substrate concentration by S, with units of mol.

The student believed that S could be approximately described by S(t) = S0e

−kt

.

The data they collected include the points S(2) = 12 and S(5) =

7.

a. How much substrate is present at t = 0?

b. How much substrate is left when t gets large? Does this depend on k?

c. What are the units of S0 and k?

d. By taking the natural log (ln) of both sides, show how ln(S)

is related to t.

e. Suppose that the student measures S at two different times, and finds that S(2) = 12 and S(5) = 7. Set up a matrix equation for ln(S0) and k.

f. Write this linear system in the form of a matrix equation, Ax =

b, and solve for S0 and k using an appropriate software package to find the inverse of A.