00:01
In the given question, we are told that suppose that $3 ,000 is placed in a savings account at an annual rate of 7 % compounded quarterly.
00:13
So we have $3 ,000 in a savings account, in a savings account, the annual interest rate, the annual interest rate, the annual interest rate is given as 7 percentage and we are told that it is compounded quarterly compounded quarterly right and now we are told that assuming that no withdrawals are made how long will it take for the account to grow to $4 ,00029 from $3 ,000 ,000 to $4 ,029.
01:06
So what we could do over here is we can use the compound interest formula, which is of the form a is equal to p times 1 plus r by n raised to the power of n times t.
01:25
So in this formula, we can just define each of these terms.
01:29
This a is the accumulated amount or we can say the amount after t years right so the accumulated amount is what a represents p is the principal amount or we can say the initial amount that is in the savings account which over here would be three thousand dollars the accumulated amount would over here would be 4 ,029 and then we have r which is the interest rate the interest rate then we have n which is the number of times interest the number of times the interest is compounded per year compounded per year over here we are told that it has compounded quarterly right which means it is compounded four times in an year so we can write n is equal to four and lastly we have time which we can take as our time in years right so now that we have defined every everything in this formula we can substitute each of the known values in this so we will have 4 ,029 as the accumulated amount the principal amount is $3 ,000 then 3 ,000 times 1 plus r is given us 7 percentage which we would write as 0 .07 so 0 .07 divided by 4 times n times t which is 4 times times t right so in the question we are told to find the time in which the $3 ,000 would grow to $4 ,029 right so the value of t is what we need to find so we can take 4 ,000 divided by 4 ,029 divided by divided by 3 ,000 and this is equal to 1 .3 4 3 and this is equal to 0 .07 divided by 4 plus 1 is 1 .0175 raised to the power of 4 times t next what we can do is let's take log on both sides of this equation right and the reason we would take log is to use a property of logarithm by which we can write if we have log of a to the power of b we can write it as b times log of a so using this property on the right hand side of the above equation what we can do is we can write log of 1 .343 is then equal to 4 times t which is the power over here times log of 1 .0175 so now now we can write then t is equal to 1 by 4 times log of 1 .343 divided by log of 1 .0175 and when we take the log of 1 .343 and divide it with the log of 1 .0175 and then multiply 1 by 4 to it.
05:25
What we would have as the answer over here is 4 .25...