practice problems corequisite n.3 name savorio s. (1) there are many types of investments. putting money into a savings account or an individual retirement account are two examples of investments. some types of investments earn interest based on a percentage rate. people often estimate the doubling time of investments to predict how much money the investment will be worth in the future. an investment that earns 5% interest will double in value about every 14 years. use this information to answer the following questions for investing $7,500 at 5% interest in 2020. be sure to explain or show your calculation work. a) in what year would the value of a $7,500 investment at 5% interest be $15,000? 15,000 is double 7,500 14 years 2020 + 14 = 2034 b) in what year would the value of a $7,500 investment at 5% interest be $30,000? c) what would the value of a $7,500 investment at 5% interest be in the year 2062? d) which of the following is the best estimate for the amount of time it would take this investment to reach a hundred thousand dollars? be sure to explain your answer. i) less than 40 years ii) between 40 and 60 years iii) between 60 and 80 years iv) more than 80 years

Question

Accepted Answer

### Part a)
# Explanation:
## Step1: Identify doubling time
The investment doubles every 14 years (since $7,500 	imes 2 = 15,000$).
## Step2: Calculate target year
Start year is 2020. Add doubling time: $2020 + 14 = 2034$.
# Answer: 2034

### Part b)
# Explanation:
## Step1: Determine number of doublings
To reach $30,000$ from $7,500$: $\frac{30,000}{7,500} = 4$, so it doubles twice.
## Step2: Calculate total years
Each doubling takes 14 years. Total years: $14 	imes 2 = 28$ years.
## Step3: Find target year
Start year 2020 + 28 years: $2020 + 28 = 2048$.
# Answer: 2048

### Part c)
# Explanation:
## Step1: Calculate years passed
From 2020 to 2062: $2062 - 2020 = 42$ years.
## Step2: Determine number of doublings
Doubling time is 14 years. Number of doublings: $\lfloor \frac{42}{14}
floor = 3$ (exact, since $14 	imes 3 = 42$).
## Step3: Calculate final value
Each doubling multiplies by 2. After 3 doublings: $7,500 	imes 2^3 = 7,500 	imes 8 = 60,000$.
# Answer: $60,000$

### Part d)
# Explanation:
## Step1: Find number of doublings needed
Target: $100,000$. Initial: $7,500$. $\frac{100,000}{7,500} \approx 13.33$. We need to find $n$ where $2^n \geq 13.33$.
## Step2: Estimate $n$
$2^3 = 8$, $2^4 = 16$ (which is $\geq 13.33$). Wait, no—wait, initial is $7,500$. Wait, let's re - express: $7,500 	imes 2^n \geq 100,000 \implies 2^n \geq \frac{100,000}{7,500} \approx 13.33$. So $n = 4$ (since $2^3 = 8$, $2^4 = 16$). Wait, no, earlier mistake: Wait, $7,500$ doubles to $15,000$ (14y), to $30,000$ (28y), to $60,000$ (42y), to $120,000$ (56y). $120,000 > 100,000$. So time is between $14 	imes 3 = 42$ and $14 	imes 4 = 56$ years. Which is between 40 and 60 years.
# Answer: ii) Between 40 and 60 years

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Part a)

Step1: Identify doubling time

Step2: Calculate target year

Step1: Determine number of doublings

Step2: Calculate total years

Step3: Find target year

Step1: Calculate years passed

Step2: Determine number of doublings

Step3: Calculate final value

Answer:

Part b)