Use the expression to solve the problems.
[tex]$P(1+r)^t$[/tex]
The amount that results when [tex]$4,000$[/tex] is compounded at [tex]$6 \%$[/tex] annually over seven years.
The interest earned in this case. [ ]
Answer (2)
Calculate the amount after 7 years using the formula: A = 4000 ( 1 + 0.06 ) 7 , which results in A = $6014.52 .
Calculate the interest earned by subtracting the principal from the amount: I = A − 4000 , which results in I = $2014.52 .
The amount that results when $4 , 000 is compounded at 6% annually over seven years is $6014.52 .
The interest earned in this case is $2014.52 .
Explanation
Understanding the Problem We are given the formula for compound interest: P ( 1 + r ) t , where:
P is the principal amount (initial investment), r is the annual interest rate (as a decimal), t is the number of years.
We are given:
P = $4 , 000 r = 6% = 0.06 t = 7 years
We need to find the amount after 7 years and the interest earned.
Calculating the Amount After 7 Years First, we calculate the amount A after 7 years using the formula:
A = P ( 1 + r ) t
Substituting the given values:
A = 4000 ( 1 + 0.06 ) 7
Amount After Compounding Now, we calculate the value of A :
A = 4000 ( 1.06 ) 7
A = 4000 × 1.503630259
A = 6014.521036
So, the amount after 7 years is approximately $6014.52 .
Calculating the Interest Earned Next, we calculate the interest earned I by subtracting the principal P from the amount A :
I = A − P
Substituting the values:
I = 6014.52 − 4000
I = 2014.52
So, the interest earned is approximately $2014.52 .
Final Answer Therefore, the amount after 7 years is $6014.52 , and the interest earned is $2014.52 .
Examples
Compound interest is a powerful tool for growing wealth over time. For example, if you invest 5 , 000 ina re t i re m e n t a cco u n tw i t hana v er a g e ann u a l re t u r n o f 8 A = P(1+r)^t a ll o w syo u t oc a l c u l a t e t h e f u t u re v a l u eo f anin v es t m e n t , w h ere A i s t h e f u t u re v a l u e , P i s t h e p r in c i p a l , r i s t h e in t eres t r a t e , an d t$ is the time in years. This concept is also crucial in understanding loans, mortgages, and other financial products.
After 7 years, the amount from compounding $4,000 at 6% annually is approximately $6014.52. The interest earned during this period is about $2014.52.
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