2. in 1985, there were 285 cell phone subscribers in the small town of centerville. the number of subscribers increased by 75% per year after 1985. in what year were there 6,209 cell phone subscribers?
3. each year the local country club sponsors a tennis tournament. play starts with 128 participants. during each round, half of the players are eliminated. how many players remain after 5 rounds?
4. the population of winnemucca, nevada, can be modeled by p = 6191(1.04)^t where t is the number of years since 1990. what was the population in 1990? by what percent did the population increase by each year?
5. you have inherited land that was purchased for $30,000 in 1960. if the value of the land was $90,000 in 1975, by what rate did the value increase by each year?
6. during normal breathing, about 12% of the air in the lungs is replaced after one breath. write an exponential decay model for the amount of the original air left in the lungs if the initial amount of air in the lungs is 500 ml. how much of the original air is present after 240 breaths?
7. you deposit $1600 in a bank account. find the rate of growth if $37,400 is in the account after 20 years.
8. you buy a new computer for $2100. the computer decreases by 50% annually. when will the computer have a value of $600?
9. the foundation of your house has about 1,200 termites. the termites grow at a rate of about 2.4% per day. how long until the number of termites doubles?
10. the half - life of cs - 137 is 30.2 years. if the initial mass of the sample is 1.00kg, how much will remain after 151 years?

Question

Accepted Answer

### 2.
# Explanation:
## Step1: Set up the exponential - growth formula
The formula for exponential growth is $A = P(1 + r)^t$, where $A$ is the final amount, $P$ is the initial amount, $r$ is the growth rate as a decimal, and $t$ is the number of years. Here, $P = 285$, $r=0.75$, and $A = 6209$. So, $6209=285(1 + 0.75)^t$.
## Step2: Solve for $t$
First, divide both sides of the equation by 285: $\frac{6209}{285}=(1.75)^t$. So, $21.786=(1.75)^t$. Take the natural - logarithm of both sides: $\ln(21.786)=t\ln(1.75)$. Then, $t=\frac{\ln(21.786)}{\ln(1.75)}$. Calculate $\ln(21.786)\approx3.084$ and $\ln(1.75)\approx0.559$. So, $t=\frac{3.084}{0.559}\approx5.52$. Since we started in 1985, the year is $1985 + 6=1991$ (we round up to the next whole year because the number of subscribers reaches 6209 during the 6th year).
# Answer:
1991

### 3.
# Explanation:
## Step1: Identify the pattern
The number of players remaining after each round follows a geometric - sequence pattern. The initial number of players $a_1 = 128$, and the common ratio $r=\frac{1}{2}$. The formula for the $n$th term of a geometric sequence is $a_n=a_1r^{n - 1}$. Here, $n = 6$ (since we start with the first round as $n = 1$ and we want to find the number of players after 5 rounds, so $n=6$).
## Step2: Calculate the number of remaining players
Substitute $a_1 = 128$, $r=\frac{1}{2}$, and $n = 6$ into the formula: $a_6=128	imes(\frac{1}{2})^{6 - 1}=128	imes(\frac{1}{2})^5$. Since $128 = 2^7$, then $a_6=2^7	imes\frac{1}{2^5}=2^{7 - 5}=4$.
# Answer:
4

### 4.
# Explanation:
## Step1: Find the population in 1990
When $t = 0$ (since $t$ is the number of years since 1990), substitute into the formula $P = 6191(1.04)^t$. So, $P(0)=6191(1.04)^0=6191$.
## Step2: Determine the percentage increase
The general form of an exponential - growth formula is $P = P_0(1 + r)^t$, where $r$ is the growth rate. Comparing $P = 6191(1.04)^t$ with $P = P_0(1 + r)^t$, we can see that $1 + r=1.04$, so $r = 0.04$ or 4%.
# Answer:
Population in 1990: 6191; Percentage increase per year: 4%

### 5.
# Explanation:
## Step1: Set up the exponential - growth formula
The formula for exponential growth is $A = P(1 + r)^t$, where $P$ is the initial value, $A$ is the final value, $r$ is the annual growth rate, and $t$ is the number of years. Here, $P = 30000$, $A = 90000$, and $t=1975 - 1960 = 15$. So, $90000=30000(1 + r)^{15}$.
## Step2: Solve for $r$
First, divide both sides of the equation by 30000: $3=(1 + r)^{15}$. Take the 15th root of both sides: $1 + r=3^{\frac{1}{15}}$. Calculate $3^{\frac{1}{15}}\approx1.07598$. Then, $r=3^{\frac{1}{15}}-1\approx0.076$ or 7.6%.
# Answer:
7.6%

### 6.
# Explanation:
## Step1: Write the exponential - decay formula
The formula for exponential decay is $A = P(1 - r)^n$, where $P$ is the initial amount, $r$ is the decay rate as a decimal, and $n$ is the number of time - intervals. Here, $P = 500$, $r = 0.12$, and the formula for the amount of original air left is $A = 500(1 - 0.12)^n=500(0.88)^n$.
## Step2: Calculate the amount of original air after 240 breaths
When $n = 240$, $A = 500(0.88)^{240}$. Using a calculator, $A\approx0$.
# Answer:
The exponential - decay model is $A = 500(0.88)^n$; Amount of original air after 240 breaths: approximately 0 mL

### 7.
# Explanation:
## Step1: Set up the exponential - growth formula
The formula for compound - growth (assuming continuous or simple compounding in a general sense) is $A = P(1 + r)^t$, where $P$ is the principal amount, $A$ is the final amount, $r$ is the annual growth rate, and $t$ is the number of years. Here, $P = 1600$, $A = 37400$, and $t = 20$. So, $37400=1600(1 + r)^{20}$.
## Step2: Solve for $r$
First, divide both sides by 1600: $\frac{37400}{1600}=(1 + r)^{20}$, so $23.375=(1 + r)^{20}$. Take the 20th root of both sides: $1 + r=23.375^{\frac{1}{20}}$. Calculate $23.375^{\frac{1}{20}}\approx1.17$. Then, $r=23.375^{\frac{1}{20}}-1\approx0.17$ or 17%.
# Answer:
17%

### 8.
# Explanation:
## Step1: Set up the exponential - decay formula
The formula for exponential decay is $A = P(1 - r)^t$, where $P$ is the initial value, $A$ is the final value, $r$ is the decay rate as a decimal, and $t$ is the number of years. Here, $P = 2100$, $r = 0.5$, and $A = 600$. So, $600=2100(1 - 0.5)^t$.
## Step2: Solve for $t$
First, divide both sides by 2100: $\frac{600}{2100}=(0.5)^t$, so $\frac{2}{7}=(0.5)^t$. Take the natural - logarithm of both sides: $\ln(\frac{2}{7})=t\ln(0.5)$. Then, $t=\frac{\ln(\frac{2}{7})}{\ln(0.5)}=\frac{\ln2-\ln7}{\ln(0.5)}$. Calculate $\ln2\approx0.693$, $\ln7\approx1.946$, and $\ln(0.5)\approx - 0.693$. So, $t=\frac{0.693 - 1.946}{-0.693}=\frac{-1.253}{-0.693}\approx1.81$. Rounding up, $t = 2$ years.
# Answer:
2 years

### 9.
# Explanation:
## Step1: Set up the exponential - growth formula
The formula for exponential growth is $A = P(1 + r)^t$, where $P$ is the initial amount, $A$ is the final amount, $r$ is the growth rate as a decimal, and $t$ is the number of time - intervals. Here, $P = 1200$, $A = 2400$ (since we want the number of termites to double), and $r = 0.024$. So, $2400=1200(1 + 0.024)^t$.
## Step2: Solve for $t$
First, divide both sides by 1200: $2=(1.024)^t$. Take the natural - logarithm of both sides: $\ln2=t\ln(1.024)$. Then, $t=\frac{\ln2}{\ln(1.024)}$. Calculate $\ln2\approx0.693$ and $\ln(1.024)\approx0.0237$. So, $t=\frac{0.693}{0.0237}\approx29.24$. Rounding up, $t\approx30$ days.
# Answer:
30 days

### 10.
# Explanation:
## Step1: Use the half - life formula
The formula for radioactive decay is $A = P(\frac{1}{2})^{\frac{t}{T_{1/2}}}$, where $P$ is the initial mass, $A$ is the final mass, $t$ is the time elapsed, and $T_{1/2}$ is the half - life. Here, $P = 1.00$ kg, $t = 151$ years, and $T_{1/2}=30.2$ years.
## Step2: Calculate the remaining mass
Substitute the values into the formula: $A = 1	imes(\frac{1}{2})^{\frac{151}{30.2}}$. Since $\frac{151}{30.2}=5$, then $A = (\frac{1}{2})^5=\frac{1}{32}=0.03125$ kg.
# Answer:
0.03125 kg

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QUESTION IMAGE

Question

Explanation:

2.

Step1: Set up the exponential - growth formula

Step2: Solve for $t$

Step1: Identify the pattern

Step2: Calculate the number of remaining players

Step1: Find the population in 1990

Step2: Determine the percentage increase

Answer:

3.