Questions: The exponential function f(x)=49.47(1.024)^x describes the population of a certain country, y, in millions, in year x. a) Substitute 0 for x and, without using a calculator, find the country's population in year 0. b) Find the country's population in year 24 as predicted by this function. A. a) 1 million; b) 91.652 million B. a) 49.47 million; b) 89.504 million C. a) 1.024 million; b) 85.357 million D. a) 49.47 million; b) 87.406 million

The exponential function f(x)=49.47(1.024)^x describes the population of a certain country, y, in millions, in year x.
a) Substitute 0 for x and, without using a calculator, find the country's population in year 0.
b) Find the country's population in year 24 as predicted by this function.
A. a) 1 million; b) 91.652 million
B. a) 49.47 million; b) 89.504 million
C. a) 1.024 million; b) 85.357 million
D. a) 49.47 million; b) 87.406 million

Transcript text: The exponential function $f(x)=49.47(1.024)^{x}$ describes the population of a certain country, $y$, in millions, in year $x$. a) Substitute 0 for $x$ and, without using a calculator, find the country's population in year 0 . b) Find the country's population in year 24 as predicted by this function. A. a) 1 million; b) 91.652 million B. a) 49.47 million; b) 89.504 million C. a) 1.024 million; b) 85.357 million D. a) 49.47 million; b) 87.406 million

Solution

Solution Steps

To solve the given problem, we need to evaluate the exponential function $ f(x) = 49.47(1.024)^x $ at specific values of $ x $.

a) For $ x = 0 $, we substitute 0 into the function and calculate the population. b) For $ x = 24 $, we substitute 24 into the function and calculate the population.

Step 1: Calculate Population in Year 0

To find the population in year 0, we substitute $ x = 0 $ into the function $ f(x) = 49.47(1.024)^x $:

\[ f(0) = 49.47(1.024)^0 = 49.47 \times 1 = 49.47 \]

Thus, the population in year 0 is $ 49.47 $ million.

Step 2: Calculate Population in Year 24

Next, we calculate the population in year 24 by substituting $ x = 24 $ into the function:

\[ f(24) = 49.47(1.024)^{24} \approx 49.47 \times 1.605 \approx 87.4059 \]

Rounding to four significant digits, the population in year 24 is approximately $ 87.41 $ million.