Can anybody show me step by step pls?
A man invests $3000 in an account that pays 7.5% interest per year, compounded quarterly.
1. Find the amount after 3 years? (Round your answer to the nearest cent.)
how much $?
2. How long will it take for the investment to double? (Round your answer to two decimal places.)
how many yrs?
i = .075/4 = .01875
1. n = 4(3) = 12
amount = 3000(1.01875)^12 = 3749.15
2.
3000(1.01875)^n = 6000 , where n is the number of quarter years
1.01875^n = 2
take logs of both sides and use use of logs
n log 1.01875 = log 2
n = log 2/log 1.01875 = appr 37.313
= 37.313../4 or appr 9.3 years
so just to clarify the answer is 37.313 or 9.3? if i was to round to two decimal places
It should be clear to you that the 37.313 is the number of quarter years, and if I divide that by 4
I would get years
37.313../4 = 9.33 years rounded to 2 decimal places
Sure, I can help you with that! To solve these questions step by step, we will use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the future amount
P = the principal amount (initial investment)
r = annual interest rate (in decimal form)
n = number of times interest is compounded per year
t = number of years
Let's solve the questions one by one:
1. Find the amount after 3 years?
Here, the principal amount (P) is $3000, the annual interest rate (r) is 7.5% (or 0.075 in decimal form), and interest is compounded quarterly (n = 4).
Substituting the given values into the formula:
A = 3000(1 + 0.075/4)^(4*3)
A = 3000(1.01875)^12
Now, calculate (1.01875)^12 using a calculator:
(1.01875)^12 ≈ 1.284
Finally, multiply the result by the principal amount (P):
A ≈ 3000 * 1.284
The future amount after 3 years is approximately $3,852.17 (rounded to the nearest cent).
2. How long will it take for the investment to double?
In this case, we want to find the value of 't' (number of years) when the future amount (A) is twice the principal amount (2P).
Using the same formula, we have:
2P = P(1 + 0.075/4)^(4t)
Dividing both sides of the equation by P:
2 = (1 + 0.075/4)^(4t)
Now, taking the natural logarithm (ln) of both sides:
ln(2) = 4t * ln(1 + 0.075/4)
Dividing both sides by 4 * ln(1.01875) gives:
t = ln(2) / (4 * ln(1.01875))
Using a calculator, calculate ln(2) and ln(1.01875):
ln(2) ≈ 0.693
ln(1.01875) ≈ 0.01857
Finally, substitute the values into the equation:
t ≈ 0.693 / (4 * 0.01857)
After evaluation, it turns out that it will take approximately 14.86 years (rounded to two decimal places) for the investment to double.
I hope this step-by-step explanation helps! Let me know if you have any further questions.