Question

The quantity of an element is decaying over time such that the amount of material

any time is given by the function F(t) = 6700(0.7)3t. Write an equivalent function
of the form F(t) = abt.

Answer 1

The decay function F(t) = 6700(0.7)3t can be written in the form F(t) = abt by rewriting the term (0.7)3t as (0.7³)ⁿ which equals 0.343, making the equivalent function F(t) = 6700(0.343)t where 'a' is 6700 and 'b' is 0.343.

Step-by-step explanation:

The student is asked to write the decay function F(t) = 6700(0.7)3t in the form F(t) = abt, which is a common format for expressing exponential decay in the context of radioactive decay or half-life problems typically found in math or science classes. To do this, we should recognize that the term (0.7)3t can be rewritten as (0.73)t, which is the same as (0.343)t because 0.73 equals 0.343. Thus the function can be written as:

F(t) = 6700(0.343)t

Here, 'a' represents the initial quantity, which is 6700, and 'b' represents the base of the exponential decay, which is 0.343 (because 0.73 equals 0.343). The decay constant can be related to this form in more advanced chemistry or physics topics.