Question

the rates of change in population for two cities are ( p(t) = 45 ) for alphaville and ( q(t) = 102e^{0.05t} ) for betaburgh, where ( t ) is the number of years since 1990, and ( p ) and ( q ) are measured in people per year. in 1990, alphaville had a population of 5000, and betaburgh had a population of 3500. answer parts a) through c). a) determine the population models for both cities. the population model for alphaville is ( p(t)=45t + 5000 ). the population model for betaburgh is ( q(t)=2040e^{0.05t}+1460 ). b) what were the populations of alphaville and betaburgh, to the nearest hundred, in 2000? the population of alphaville in 2000 was ( square ) people. (round to the nearest hundred as needed)

Explanation:

Step1: Find the value of \( t \)

Since \( t \) is the number of years since 1990, for the year 2000, \( t = 2000 - 1990=10 \)

Step2: Calculate the population of Alphaville

We use the population model \( P(t)=45t + 5000 \). Substitute \( t = 10 \) into the formula:
\( P(10)=45\times10+5000=450 + 5000=5450 \)

Step3: Round to the nearest hundred

To round 5450 to the nearest hundred, we look at the tens digit. The tens digit is 5. Using the rounding rule (\( \geq5 \) round up), \( 5450\approx5500 \)

Answer:

5500