which of the following functions represents a population that is decreasing at a rate of 1.25% each year? a. $f(x) = 7,000(1.125)^x$ b. $f(x) = 7,000(1.0125)^x$ c. $f(x) = 7,000(0.875)^x$ d. $f(x) = 7,000(0.9875)^x$ a small town has a current population of 20,000 people. due to new industry, the population is expected to grow at a rate of 1.8% every year. 1. identify the a - value and the b - value for this scenario. 2. write an exponential function, $p(t)$, to represent the towns population after t years. a scientist is studying two different bacteria cultures. - culture a: $f(x)=100(1 + 0.05)^x$ - culture b: $g(x)=500(1 - 0.02)^x$ 1. describe the behavior of each culture (growth or decay) and identify the percentage rate of change for each. 2. justify: even though culture b starts with a higher initial population, explain why culture a will eventually have more bacteria than culture b.

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Accepted Answer

### First Question (Which function represents a population decreasing at 1.25% yearly?)
# Explanation:
## Step1: Recall exponential decay formula
The general form for exponential decay is $ f(x) = a(1 - r)^x $, where $ a $ is the initial amount, $ r $ is the rate of decrease (in decimal), and $ x $ is time. For a decrease of 1.25% (or $ r = 0.0125 $), the base is $ 1 - 0.0125 = 0.9875 $.
## Step2: Analyze each option
- Option A: $ 1.125>1 $, so it's growth (12.5% growth), not decay.
- Option B: $ 1.0125>1 $, growth (1.25% growth), not decay.
- Option C: $ 0.875 = 1 - 0.125 $, which would be 12.5% decay, not 1.25%.
- Option D: $ 0.9875 = 1 - 0.0125 $, which is 1.25% decay.
# Answer: D. $ f(x) = 7,000(0.9875)^x $

### Second Question (Small town population)
#### 1. Identify $ a $-value and $ b $-value
# Explanation:
## Step1: Recall exponential growth formula
The general form for exponential growth is $ P(t) = a(1 + r)^t $, where $ a $ is the initial population, $ r $ is the growth rate (in decimal), and $ t $ is time. The base $ b = 1 + r $.
## Step2: Determine $ a $ and $ b $
- Initial population $ a = 20,000 $ (current population).
- Growth rate $ r = 1.8\% = 0.018 $, so $ b = 1 + 0.018 = 1.018 $.
# Answer:
$ a $-value: $ 20,000 $
$ b $-value: $ 1.018 $

#### 2. Write the exponential function $ P(t) $
# Explanation:
## Step1: Use the exponential growth formula
Substitute $ a = 20,000 $ and $ b = 1.018 $ into $ P(t) = a \cdot b^t $.
## Step2: Form the function
$ P(t) = 20,000(1.018)^t $
# Answer: $ P(t) = 20,000(1.018)^t $

### Third Question (Bacteria cultures)
#### 1. Describe behavior and percentage rate of change
# Explanation:
## Step1: Analyze Culture A ($ f(x) = 100(1 + 0.05)^x $)
- The form $ a(1 + r)^x $ indicates growth (since $ 1 + r>1 $). The rate $ r = 0.05 = 5\% $, so Culture A is growing at 5% per time unit.
## Step2: Analyze Culture B ($ g(x) = 500(1 - 0.02)^x $)
- The form $ a(1 - r)^x $ indicates decay (since $ 1 - r<1 $). The rate $ r = 0.02 = 2\% $, so Culture B is decaying at 2% per time unit.
# Answer:
- Culture A: Growth, 5% rate of change (growth rate).
- Culture B: Decay, 2% rate of change (decay rate).

#### 2. Justify why Culture A will eventually have more bacteria than Culture B
# Explanation:
## Step1: Compare growth/decay rates
Culture A is growing (5% per unit time), so its population increases over time. Culture B is decaying (2% per unit time), so its population decreases over time. Even though Culture B starts with a higher initial population (500 vs. 100 for Culture A), the growth of Culture A (increasing by 5% of its current population) will eventually outpace the decay of Culture B (decreasing by 2% of its current population). Over enough time, the increasing population of A will surpass the decreasing population of B.
# Answer: Culture A grows (5% rate) while Culture B decays (2%). Since A’s population increases and B’s decreases, A’s growth will eventually make its population exceed B’s, even with B’s higher initial count.

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QUESTION IMAGE

Question

Explanation:

First Question (Which function represents a population decreasing at 1.25% yearly?)

Step1: Recall exponential decay formula

Step2: Analyze each option

Step1: Recall exponential growth formula

Step2: Determine \( a \) and \( b \)

Step1: Use the exponential growth formula

Step2: Form the function

Answer:

Second Question (Small town population)

1. Identify \( a \)-value and \( b \)-value