A national census bureau predicts that a certain population will increase from 38.1 million in 2000 to 55.5 million in 2080. Complete parts (a) through (c) below.

(a) Find an exponential function of the form \( f(t) = y_0b^t \) for these data, in which \( t = 0 \) corresponds to 2000 and \( f(t) \) is in millions.

\( f(t) = \)

(Use integers or decimals for any numbers in the expression. Round to four decimal places as needed.)

To find the exponential function representing the population growth, we use the formula f(t) = y0 * b^t, where t is the number of years after 2000 and y0 is the initial population in 2000. The population growth from 2000 to 2080 is represented by the exponential function f(t) = 38.1 * (55.5/38.1)^t.

Explanation:

To find an exponential function that represents the population growth from 2000 to 2080, we can use the formula f(t) = y0 * b^t. Here, t represents the number of years after 2000, y0 is the initial population in 2000, and b is the growth factor. In this case, the initial population is 38.1 million in 2000 (t=0), and the population in 2080 (t=2080-2000=80) is 55.5 million. Plugging these values into the formula, we get f(t) = 38.1 * (55.5/38.1)^t. This gives us the exponential function that represents the population growth.

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