Questions: Caladan is currently one of the world's fastest-growing countries. The exponential function f(x)=55(1.034)^x models the population of Caladan, f(x), in millions, x years after 1974. Using this exponential function and a calculator with a y* key or a ^ key, answer the following questions. a. Substitute 0 for x and, without using a calculator, find Caladan's population in 1974. 55 million b. Substitute 21 for x and use your calculator to find Caladan's population in the year 1995 as predicted by this function. million (Round to the nearest tenth.)

Solution Steps

Substitute \( x = 0 \) into the exponential function \( f(x) = 55(1.034)^{x} \): \[ f(0) = 55(1.034)^{0}. \] Since any number raised to the power of 0 is 1, this simplifies to: \[ f(0) = 55 \times 1 = 55. \] Thus, Caladan's population in 1974 was 55 million.

Substitute \( x = 21 \) into the exponential function \( f(x) = 55(1.034)^{x} \): \[ f(21) = 55(1.034)^{21}. \]

Use a calculator to compute \( (1.034)^{21} \): \[ (1.034)^{21} \approx 2.039. \]

Multiply the result by 55 to find the population in 1995: \[ f(21) = 55 \times 2.039 \approx 112.145. \] Round to the nearest tenth: \[ f(21) \approx 112.1. \] Thus, Caladan's population in 1995 was approximately 112.1 million.

Final Answer