anonymous
  • anonymous
Rachel deposited $5,960.32 into a savings account with an interest rate of 4.2% compounded twice a year. About how long will it take for the account to be worth $9,000? a. 21 years, 1 month b. 18 years, 0 months c. 19 years, 10 months d. 9 years, 11 months
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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kropot72
  • kropot72
\[A=P(1+\frac{r}{n})^{nt}\] where A is the amount after t years P is the principal r is the annual interest rate as a decimal (0.042 in this case) n is the number of times per year that the interest compounded Plugging in the given values we get: \[9000=5960.32(1+\frac{0.042}{2})^{2t}\] Now you need to solve for t.
kropot72
  • kropot72
@xoxo_devvie Are you there?
anonymous
  • anonymous
sorry about that. thank you so much! you show me the process? im taking online classes and dont have a teacher here with me and im trying to learn the lesson by myself

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kropot72
  • kropot72
np. Do you need help to solve?
anonymous
  • anonymous
yes please
kropot72
  • kropot72
\[9000=5960.32(1+\frac{0.042}{2})^{2t}\] First divide both sides by 5960.32, and add the terms inside brackets giving: \[\frac{9000}{5960.32}=(1.021)^{2t}\] Next take natural logs of both sides to get: \[\ln 1.509986=2t \times \ln 1.021\] Now divide both sides by ln 1.021 getting \[2t=\frac{\ln 1.509981}{\ln 1.021}\] Then dividing both sides by 2 we finally get: \[t=\frac{\frac{\ln 1.509981}{\ln 1.021}}{2}\]
anonymous
  • anonymous
how do i know how to convert it into years and months to find the answer?
kropot72
  • kropot72
First calculate t as a decimal quantity. The years do not need conversion.Then we can convert the decimal fraction to months.
kropot72
  • kropot72
Use your calculator to find the value of \[\frac{\ln 1.509986}{\ln 1.021}\] Then divide the result by 2 to get the value of t. Can you do that?
anonymous
  • anonymous
one second let me see
anonymous
  • anonymous
i got 0.739?...
kropot72
  • kropot72
Did you use the natural logs function on your calculator. It is the 'ln' function.
anonymous
  • anonymous
i only have a basic calculator :(
kropot72
  • kropot72
The is a free online calculator here: http://web2.0calc.com/ Put the cursor on the 'log' key and select 'ln' on the drop-down menu.
anonymous
  • anonymous
ok!
anonymous
  • anonymous
.391 is what i got this time?
kropot72
  • kropot72
Well I used the online calculator and got a correct result. Can we go through using it step by step?
anonymous
  • anonymous
yes!
kropot72
  • kropot72
Put the cursor on the 'log' key and select 'ln' on the drop down menu. What do you see in the window at the top of the calculator?
anonymous
  • anonymous
this is what i see
1 Attachment
kropot72
  • kropot72
Great! Now enter 1.509981, followed by a closing bracket. Next click on the division key and then put the cursor on the 'log' key again and select 'ln'. Enter 1.021, followed by a closing bracket. Then click on the '=' key.
anonymous
  • anonymous
so the answer is 3?
kropot72
  • kropot72
No. When I do the operations the result is 19.8290047403184753. When this is divided by 2 the result is 9.91450237015923765. This converts to 9 years, 11 months.
anonymous
  • anonymous
how did you get that ?
anonymous
  • anonymous
oh! there we go i got it! do you think you can help me with one more?
kropot72
  • kropot72
Good work! Sorry I have to go now. I am sure others can help :)

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