choose the best answer from the four choices given. answers are on page 281.
1. the number of rabbits in a certain population doubles every 3 months. currently, there are 5 rabbits in the population. how many rabbits will there be 3 years from now?
a. 64
b. 4,096
c. 9,064
d. 20,480
2. hannah took out a simple interest loan of $1,000. the interest rate on the loan is 5%. if no payments are made, how much money, in dollars, will she owe at the end of 7 years?
a. (1.05)^7
b. 7(1.05)
c. 1,000 + (1,000)(0.05)^7
d. 1,000 + 7(0.05)(1,000)
3. the number of bacteria in a certain culture doubles every 15 minutes. if the number was initially 10^3, what was the number in the population 1 hour later?
a. (10^3)2^(15/60)
b. (10^3)4^(15/60)
c. 2^(60/15)(10^3)
d. 2^3(10)(60/15)
4. during a two - year period of political instability in a country, the price of a \market basket\ of essential goods doubled every six months from its base price of 1 wint, the unit of currency of the country. which of the lines on the graph below best represents the rise in the price of the market basket?
graph: price of market basket, x - axis: 0 mos., 6 mos., 12 mos., 18 mos., 24 mos.; y - axis: price in wints (0 - 24). lines: a (steep straight), b (shallow straight), c (curved up), d (moderate straight), e (very shallow straight)
a. a
b. b
c. c
d. d
5. riley invests $12,000 in an account that pays 6.5% interest per year compounded quarterly. to the nearest dollar, what is the total amount of money he will have after 4 years?
a. $3,531
b. $12,000
c. $13,530
d. $15,531

Question

Accepted Answer

### Question 1
# Explanation:
## Step1: Determine the number of doubling periods.
3 years is equal to $ 3 \times 12 = 36 $ months. Each doubling period is 3 months, so the number of periods $ n = \frac{36}{3}=12 $.
## Step2: Use the exponential growth formula.
The formula for exponential growth when doubling is $ N = N_0 \times 2^n $, where $ N_0 = 5 $ (initial number of rabbits) and $ n = 12 $. So $ N = 5\times2^{12} $.
Calculate $ 2^{12}=4096 $, then $ 5\times4096 = 20480 $? Wait, no, wait: Wait, $ 2^{12}=4096 $, $ 5\times4096 = 20480 $? But let's check again. Wait, 3 years is 36 months, 36 divided by 3 is 12. So initial is 5, after 12 doublings: $ 5\times2^{12}=5\times4096 = 20480 $? But the options have D as 20,480. Wait, but let's check the options again. Wait, maybe I made a mistake. Wait, 3 years is 12 quarters of 3 months. Wait, 5 rabbits, double every 3 months. So after 1 period (3 months): 10, 2 periods:20, 3:40, 4:80, 5:160, 6:320, 7:640, 8:1280, 9:2560, 10:5120, 11:10240, 12:20480. Yes, so the answer is D.
# Answer: D. 20,480

### Question 2
# Explanation:
## Step1: Recall the simple interest formula.
The formula for simple interest is $ A = P(1 + rt) $, where $ P $ is the principal amount, $ r $ is the annual interest rate (in decimal), and $ t $ is the time in years.
Here, $ P = 1000 $, $ r = 0.05 $, and $ t = 7 $.
## Step2: Substitute the values into the formula.
$ A = 1000+1000\times0.05\times7=1000 + 7\times0.05\times1000 $, which matches option D.
# Answer: D. $ 1,000 + 7(0.05)(1,000) $

### Question 3
# Explanation:
## Step1: Determine the number of doubling periods.
1 hour is 60 minutes. Each doubling period is 15 minutes, so the number of periods $ n=\frac{60}{15} = 4 $. Wait, no: Wait, the formula for exponential growth when doubling is $ N = N_0\times2^n $, where $ n $ is the number of doubling periods. The initial number $ N_0 = 10^3 $. The number of 15 - minute periods in 1 hour (60 minutes) is $ \frac{60}{15}=4 $. So $ N = 10^3\times2^4 $. But let's look at the options. Option C is $ 2^{(60/15)}(10^3)=2^4\times10^3 $, which is correct. Let's check the options:
Option A: $ (10^3)2^{(15/60)} $ is wrong. Option B: $ (10^3)2^{(15/60)} $ same as A. Option C: $ 2^{(60/15)}(10^3)=2^4\times10^3 $, correct. Option D: $ 2^3(10)(60/15) $ is wrong. So the answer is C.
# Answer: C. $ 2^{(60/15)}(10^3) $

### Question 4
# Explanation:
The price doubles every 6 months. Let's track the price over time:
- At 0 months: price = 1 wint.
- At 6 months: price = $ 1\times2 = 2 $ wints.
- At 12 months: price = $ 2\times2=4 $ wints.
- At 18 months: price = $ 4\times2 = 8 $ wints.
- At 24 months: price = $ 8\times2=16 $ wints.

Now let's check the graph:
- Line A: At 24 months, it's around 24, too high.
- Line B: It's a curve, but our growth is exponential (doubling every 6 months), so it should be a curve? Wait, no, the options: Wait, the graph has lines A (straight line from 0 to 24, slope high), D (straight line), C (curve), B (curve). Wait, our price at 6 months:2, 12:4, 18:8, 24:16. So let's see the points:
At 6 months (0.5 years), price 2; 12 months, 4; 18 months, 8; 24 months, 16. So the points are (0,1), (6,2), (12,4), (18,8), (24,16). This is an exponential curve, and among the options, line C or B? Wait, let's check the graph: Line C: at 24 months, around 16? Wait, the graph: the y - axis is price in wints, x - axis is months. At 24 months, line C reaches around 16, line B is lower, line D is a straight line, line A is a straight line with higher slope. So the correct line is C? Wait, no, let's check the options. The options are A, B, C, D. Wait, the price doubles every 6 months, so the sequence is 1,2,4,8,16 at 0,6,12,18,24 months. So the graph that passes through (0,1), (6,2), (12,4), (18,8), (24,16) is a curve, and among the options, line C? Wait, the option C is a curve that goes from (0,0) to (24,16)? No, wait, the base price is 1 wint, so at 0 months, price is 1, not 0. Wait, the graph: the lines: Line A starts at (0,4) maybe? Wait, the graph shows: at 0 mos, some lines start at 0, some at 4. Wait, the problem says "from its base price of 1 wint". So at 0 months, price is 1. So the line that starts at (0,1) and doubles every 6 months. Looking at the graph, line C: at 0 mos, it's at 0? No, maybe I misread. Wait, the options: A is a straight line from (0,4) to (24,24+), B is a curve from (0,0) to (24,10+), C is a curve from (0,0) to (24,16), D is a straight line from (0,0) to (24,16). Wait, no, the price starts at 1, so at 0 months, price is 1. So the line that has (0,1), (6,2), (12,4), (18,8), (24,16) is a curve, and among the options, line C? Wait, the answer is C? Wait, the option C is the curve that matches the exponential growth. So the answer is C.
# Answer: C. C

### Question 5
# Explanation:
## Step1: Recall the compound interest formula.
The formula for compound interest compounded quarterly is $ A = P(1+\frac{r}{n})^{nt} $, where $ P=\$12000 $ (principal), $ r = 0.065 $ (annual interest rate), $ n = 4 $ (number of times compounded per year), and $ t = 4 $ (number of years).
## Step2: Substitute the values into the formula.
First, calculate $ \frac{r}{n}=\frac{0.065}{4}=0.01625 $, and $ nt=4\times4 = 16 $.
Then $ A = 12000\times(1 + 0.01625)^{16} $.
Calculate $ (1 + 0.01625)^{16}\approx(1.01625)^{16} $. We know that $ (1 + 0.01625)^{16}\approx1.2942 $ (using a calculator: $ 1.01625^{16}\approx e^{16\times\ln(1.01625)}\approx e^{16\times0.01613}\approx e^{0.2581}\approx1.294 $).
Then $ A = 12000\times1.2942\approx15530.4 $, which is approximately \$15,531.
# Answer: D. \$15,531

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QUESTION IMAGE

Question

Explanation:

Question 1

Step1: Determine the number of doubling periods.

Step2: Use the exponential growth formula.

Step1: Recall the simple interest formula.

Step2: Substitute the values into the formula.

Step1: Determine the number of doubling periods.

Answer:

Question 2