question 1- The population of a town in 2005 was 16,500. The population has increased at a rate of 12% per year since then. graph the population growth over time since 2005.
question 2- Nathan wants to put $500 into a savings account. If he puts the money into an account at a local credit union it will earn 2% interest per month. The function below represents how much money will be in the account if he puts the money into an account for x months at a local bank.
g(x) = 500(1.015)x
After 1 year, which account would have more money in it and approximately how much more money would be in that account?
THANK YOUUU
y = 16500 * 1.12^x
You know that all such exponential graphs (y = a*b^x) have a horizontal asymptote of y=0 and go through (0,a)
(1 + 0.02/12)^12 = 1.020184 > 1.015
To answer the first question:
Step 1: Determine the population in each year since 2005.
- Start with the initial population in 2005: 16,500.
- Calculate the population for each subsequent year by increasing it by 12% annually. For example, to find the population in 2006, multiply 16,500 by (1 + 0.12).
- Repeat this calculation for each year until you reach the desired time period.
Here is a table demonstrating the population growth over time since 2005:
| Year | Population |
|-------|--------------|
| 2005 | 16,500 |
| 2006 | 18,480 |
| 2007 | 20,659.2|
| 2008 | 23,143.8|
| 2009 | 25,959.9|
| 2010 | 29,137.8|
| 2011 | 32,710.9|
| 2012 | 36,716.1|
| 2013 | 41,195.7|
| 2014 | 46,197.5|
To better visualize the population growth over time, you can plot these data points on a graph with Year (x-axis) and Population (y-axis).
For the second question:
Step 1: Compare the two functions to determine which account would have more money after 1 year.
- The function representing the amount of money in the account at the credit union is not provided, but assuming it earns interest monthly, a similar function can be constructed: f(x) = 500(1 + 0.02)^12x. Note that 2% interest per month is equivalent to a monthly interest rate of 0.02 and the exponent of 12x accounts for 12 months in a year.
- Substitute x = 1 into both functions and evaluate them.
- For g(1) = 500(1.015)^1, calculate the result.
- For f(1) = 500(1.02)^12, calculate the result.
- Compare the values obtained from both calculations.
The account with the greater value after 1 year would have more money in it. To determine the approximate difference in the amounts, you can subtract the value of the smaller account from the value of the larger account.
Remember to evaluate g(x) and f(x) using a calculator or programming software to obtain precise values.
To graph the population growth over time since 2005:
Step 1: Define the variables:
Let t represent the number of years since 2005.
Let P(t) represent the population at time t.
Step 2: Find the population for each year:
The initial population in 2005 is given as 16,500.
Since the population increases by 12% each year, we can calculate the population for year t using the formula:
P(t) = 16,500 * (1 + 0.12)^t
Step 3: Create a table of values:
Choose some values for t, such as 0, 1, 2, 3, etc., and calculate the corresponding population using the formula from step 2. For example:
t = 0 (2005): P(0) = 16,500 * (1 + 0.12)^0 = 16,500
t = 1 (2006): P(1) = 16,500 * (1 + 0.12)^1
t = 2 (2007): P(2) = 16,500 * (1 + 0.12)^2
t = 3 (2008): P(3) = 16,500 * (1 + 0.12)^3
Step 4: Plot the points on a graph:
Represent years on the x-axis and population on the y-axis. Plot the values from the table on the graph. Connect the points with a smooth curve to represent the population growth over time since 2005.
To answer question 2:
For Nathan's savings account, the function g(x) = 500(1.015)^x represents how much money will be in the account after x months with a 2% monthly interest rate.
To find out which account would have more money after 1 year (12 months), we can substitute x = 12 in the function:
g(12) = 500(1.015)^12
Calculate this value using a calculator or by hand to get the amount of money in the account after 1 year.
To compare the two accounts, subtract the amount in Nathan's bank account from the amount in the credit union account:
Difference = Amount in credit union account - Amount in Nathan's bank account
This will give you the approximate amount of money that would be in the credit union account compared to Nathan's bank account after 1 year.