Richey Loaner inherited $2105.22 from a not-so-rich uncle. If Richie deposits his money into an account that pays 3% compounded continuously for t years, then the function A(t)=2105.22e^0.03t gives the account balance after t years.
(a) Find Richie’s account balance 6 years from now.
(b) Show all the work to algebraically find the number of years it will take for Richie
to become a millionaire ($1,000,000)? Show all steps to solve algebraically.
you have the formula. Just plug in t=6
To find how long it takes to get to a million,
2105.22e^0.03t = 1000000
e^.03t = 475
.03t = ln 475 = 6.1633
t = 205.44 years
Looks like 3% isn't gonna cut it, especially if you include inflation.
y=x+3
y=4x
To find Richie's account balance 6 years from now, we can use the formula A(t) = 2105.22e^(0.03t).
(a) For t = 6, we substitute the value into the equation:
A(6) = 2105.22e^(0.03*6)
Calculating this, we have:
A(6) = 2105.22e^(0.18)
Using a calculator, we find that e^(0.18) ≈ 1.1977. Multiplying this by 2105.22, we get:
A(6) ≈ 2105.22 * 1.1977 ≈ 2518.81
So Richie's account balance 6 years from now would be approximately $2518.81.
(b) To find the number of years it will take for Richie to become a millionaire ($1,000,000), we need to solve the equation A(t) = 1000000.
Substituting the values into the equation, we get:
1000000 = 2105.22e^(0.03t)
Dividing both sides of the equation by 2105.22, we have:
e^(0.03t) = 1000000 / 2105.22
Now we need to isolate e^(0.03t). To do this, we take the natural logarithm (ln) of both sides of the equation:
ln(e^(0.03t)) = ln(1000000 / 2105.22)
The natural logarithm and the exponential function e^(0.03t) are inverse functions, so they cancel out:
0.03t = ln(1000000 / 2105.22)
Now we can solve for t by dividing both sides of the equation by 0.03:
t = ln(1000000 / 2105.22) / 0.03
Using a calculator, we find that ln(1000000 / 2105.22) / 0.03 ≈ 52.7
This means it will take approximately 52.7 years for Richie to become a millionaire ($1,000,000).