Question

Jacob and Jessica are studying the spread of

dandelions. Jacob discovers that the growth over t
weeks can be defined by the function f(t) = (8) 2¹.
Jessica finds that the growth function over t weeks
is g(t) = 2¹+³. Calculate the number of dandelions
that Jacob and Jessica will each have after 5 weeks.
Based on the growth from both functions, explain
the relationship between f(t) and g(t).

Answer 1

Explanation:

To calculate the number of dandelions that Jacob and Jessica will each have after 5 weeks, we need to evaluate their respective growth functions at t = 5.

For Jacob, f(5) = (8)2^5 = 256 dandelions after 5 weeks.

For Jessica, g(5) = 2^(1+3) = 2^4 = 16 dandelions after 5 weeks.

Therefore, Jacob will have 256 dandelions and Jessica will have 16 dandelions after 5 weeks.

To explain the relationship between f(t) and g(t), we can simplify each function:

f(t) = (8)2^t = 2^(3+t)

g(t) = 2^(1+3t)

We can see that both functions are exponential functions with a base of 2. However, the exponent in f(t) is (3+t), while the exponent in g(t) is (1+3t). This means that the two functions have different rates of growth.

Specifically, the function f(t) has a constant rate of growth, as the exponent is linearly increasing with time. On the other hand, the function g(t) has an increasing rate of growth, as the exponent is growing at a rate of 3 units per week. This means that, as time goes on, the difference in the number of dandelions between the two functions will become larger and larger.