Anne now has a balance of $900 on her credit card, on which 1.5% interest per month is charged. Assume that she makes no further purchases or payments (and that the credit card company doesn't turn her account over to a bill collector).
(a) Find the rule of the function g that gives Anne's total credit card debt after x months.
g(x) =
b)How much will Anne owe after one year? (Round your answer to the nearest cent.)
How much will Anne owe after two years? (Round your answer to the nearest cent.)
(c) When will she owe twice the amount she owes now? (Round your answer to nearest month.)
After months
(a) To find the rule of the function g that gives Anne's total credit card debt after x months, we need to calculate the interest accumulated each month and add it to the previous balance.
The monthly interest is given as 1.5% of the current balance, so it can be expressed as 0.015 times the current balance.
The rule of the function g can be written as:
g(x) = previous balance + interest
g(x) = (previous balance) + (0.015 * previous balance)
g(x) = (1 + 0.015) * previous balance
g(x) = 1.015 * previous balance
(b) After one year (12 months), we can calculate the total credit card debt using the function g:
g(12) = 1.015 * $900
g(12) ≈ $914.85
After one year, Anne will owe approximately $914.85.
To find out how much Anne will owe after two years (24 months), we can again use the function g:
g(24) = 1.015 * $914.85
g(24) ≈ $930.63
After two years, Anne will owe approximately $930.63.
(c) To find when Anne will owe twice the amount she owes currently ($900), we need to solve the equation:
2 * $900 = 1.015 * previous balance
Simplifying the equation, we get:
$1800 = 1.015 * previous balance
Dividing by 1.015 on both sides:
previous balance = $1800 / 1.015
previous balance ≈ $1773.40
Therefore, Anne will owe twice the amount she owes now after approximately 1 month (rounded to the nearest month).
To find the rule of the function g that gives Anne's total credit card debt after x months, we can use the formula for compound interest.
The formula for compound interest is: A = P(1 + r)^n
Where:
A = the final amount
P = the principal amount (initial balance)
r = interest rate per period
n = number of periods
In this case, the principal amount (initial balance) is $900, the interest rate per month is 1.5% (or 0.015 as a decimal), and the number of periods is x months. Therefore, the function g(x) can be expressed as:
g(x) = 900(1 + 0.015)^x
To find how much Anne will owe after one year (12 months), we can plug in x = 12 into the function g(x):
g(12) = 900(1 + 0.015)^12
To find how much Anne will owe after two years (24 months), we can plug in x = 24 into the function g(x):
g(24) = 900(1 + 0.015)^24
To find when Anne will owe twice the amount she owes now, we need to solve the equation:
2 * 900 = 900(1 + 0.015)^x
To do this, we can divide both sides of the equation by 900 and then take the logarithm of both sides:
2 = (1 + 0.015)^x
log2 = log((1 + 0.015)^x)
Using logarithm properties, we can bring down the exponent x:
log2 = x * log(1 + 0.015)
Finally, we can solve for x by dividing both sides of the equation by log(1 + 0.015):
x = log2 / log(1 + 0.015)
Once we have the value of x, we can round it to the nearest month to determine when Anne will owe twice the amount she owes now.
Pt = Po + r*t*Po.
b. Pt = 900 + 0.015*12*900,
= 900 + 162 = 1062.
Pt = 900 + 0.015*24*900,
= 900 + 324 = 1224.
c. Pt = 900 + 0.015*t*900 = 2*1224,
900 + 13.5t = 2448,
13.5t = 2448 - 900,
13.5t = 1548,
t = 1548 / 13.5 = 115 months.