Question

The quantity of an element is decaying over time such that the amount of material at any time is given by the function (F(t) = 400(0.7)^{3t}). Write an equivalent function of the form (F(t) = ab^t).

a. (F(t) = 400 ⋅ 0.7 ⋅ t)
b. (F(t) = 400 ⋅ (0.7)^t)
c. (F(t) = 400 ⋅ (0.7)^{3t})
d. (F(t) = 400t ⋅ (0.7)^3)

Answer 1

The equivalent function of F(t) = 400(0.7)^{3t} in the form F(t) = ab^t is c. F(t) = 400 · (0.7)^{3t}, where 'a' is 400 and 'b' is (0.7)^3.

Step-by-step explanation:

The student is asking how to write the function F(t) = 400(0.7)^{3t} in the form F(t) = ab^t, where 'a' is the initial amount and 'b' is the common ratio. To express this function correctly, we should recognize that 'a' is the initial quantity, and 'b' is the base raised to the exponent that includes the variable 't'.

The original function is F(t) = 400(0.7)^{3t}. The coefficient '400' is already in the correct place to be 'a', and '0.7' is the base for 'b'. However, because 'b' should be raised to the power of 't', we need to include the entire exponent of '3t' with the base, meaning 'b' would be '0.7'^3. So the equivalent function, where 'b' is raised to the 't' power, is F(t) = 400(0.7)^{3t}.

Therefore, the correct answer is c. F(t) = 400 · (0.7)^{3t}