the functions shown represent the populations of six cities after x years. which function best represents each situation.
The initial population of this city is 15,000
The population of this city decreases 5% each year
The population of this city increases its 15% each year.
A) f(X)= 14,000(0,05)^X
B) G(X)= 17,000(0.15)^x
C) K(X)= 15,000(0.85)^x
D) P(X)=10,000(0.95)^x
E) R(X)=18,000(1.05)^x
F) T(X)=12,000(1.15)^x
To identify which function best represents each situation, we need to match the initial populations and growth rates of the cities with the functions provided.
1. The initial population of this city is 15,000.
Both C) and E) have an initial population of 15,000, so we need to look at the growth rates to determine the correct function.
2. The population of this city decreases 5% each year.
A decreasing population means we are looking for a function with a base less than 1 (for decay). The 5% decrease can be represented as a 95% retention rate each year, or 0.95.
3. The population of this city increases by 15% each year.
An increasing population means we are looking for a function with a base greater than 1 (for growth). The 15% increase can be represented as a 115% growth rate each year, or 1.15.
Let's match these with the functions provided:
A) \( f(x) = 14,000(0.05)^x \)
- Incorrect for all situations because the initial population doesn't match any given scenario.
B) \( g(x) = 17,000(0.15)^x \)
- Incorrect for all situations because the initial population doesn't match any given scenario and the growth rate indicates a very steep decrease, which is not one of the scenarios.
C) \( k(x) = 15,000(0.85)^x \)
- This represents a decline from the initial population of 15,000. However, the rate of decrease is 15%, not 5%, therefore this is not a match with any given situation.
D) \( p(x) = 10,000(0.95)^x \)
- Incorrect for all situations because the initial population doesn't match any given scenario, though the rate would correctly represent a 5% annual decrease if the initial population matched.
E) \( r(x) = 18,000(1.05)^x \)
- Incorrect for all situations because the initial population doesn't match any given scenario, though the rate would correctly represent a 5% annual increase if the initial population matched.
F) \( t(x) = 12,000(1.15)^x \)
- Incorrect for all situations because the initial population doesn't match any given scenario, though the rate would correctly represent a 15% annual increase if the initial population matched.
None of the functions given perfectly match the descriptons provided for the situation of the cities. There appears to be an inconsistency or error in the options provided. Let's examine what would be correct for each situation:
- The initial population of a city is 15,000 and decreases 5% each year would be given by a function like \( 15,000(0.95)^x \), which is not provided in the options.
- The population of a city that starts at 15,000 and increases 15% each year would have a function like \( 15,000(1.15)^x \), which is also not provided in the options.
Given this, scenarios 2 and 3 do not have functions listed that accurately represent the situations provided. However, if this is an exercise and you must choose from the provided options, you would perhaps select the closest matches, which are:
2. The population of this city decreases 5% each year: None match exactly, but D) \( p(x) = 10,000(0.95)^x \) is the closest in terms of growth rate but not initial population.
3. The population of this city increases by 15% each year: None match exactly, but F) \( t(x) = 12,000(1.15)^x \) is the closest in terms of growth rate but not initial population.