An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (2)
Calculate the semiannual interest rate: 6%/2 = 3% = 0.03 .
Calculate the number of compounding periods: 10 × 2 = 20 .
Use the future value formula: F V = P V ( 1 + r ) n = 5000 ( 1 + 0.03 ) 20 .
Calculate the future value: F V = 5000 × 1.8061112 = 9030.556 , so the final answer is $9 , 030.56 .
Explanation
Understanding the Problem We are asked to find the amount to which $5,000 would grow in ten years at 6% compounded semiannually. We are given a table of values and four possible answers: $15,128.00, $6,719.58, $9,030.56, and $8,954.24.
Identifying Key Information The principal amount is $5,000, the annual interest rate is 6%, and the time period is 10 years. Since the interest is compounded semiannually, the interest rate per compounding period is 6%/2 = 3%, and the number of compounding periods is 10*2 = 20.
Recalling the Future Value Formula The future value of an investment compounded periodically is given by the formula: F V = P V ( 1 + r ) n where:
FV is the future value of the investment
PV is the present value (principal amount)
r is the interest rate per compounding period
n is the number of compounding periods
Calculating the Future Value In this case, PV = 5 , 000 , r = 0.03 , an d n = 20. Pl ugg in g t h ese v a l u es in t o t h e f or m u l a , w e g e t : F V = 5000 ( 1 + 0.03 ) 20 = 5000 ( 1.03 ) 20 C a l c u l a t in g t hi s v a l u e : F V = 5000 × 1.8061112 = 9030.556 $
So, the future value is approximately $9,030.56.
Selecting the Correct Answer Comparing the calculated future value with the given options, we see that the closest answer is $9,030.56.
Final Answer Therefore, the amount to which $5,000 would grow in ten years at 6% compounded semiannually is $9,030.56.
Examples
Understanding compound interest is crucial for making informed financial decisions. For instance, when planning for retirement, knowing how your savings will grow over time with compound interest helps you estimate the funds you'll have available. Similarly, when taking out a loan, understanding the compound interest can help you assess the total cost of borrowing and compare different loan options effectively. Compound interest impacts various aspects of personal finance, from investments to debt management.
A current of 15.0 A for 30 seconds results in approximately 450 C of charge flowing through the device. This charge corresponds to about 2.81 billion billion electrons, calculated using the relationship between current, charge, and the charge of a single electron. Thus, around 2.81 × 10²¹ electrons flow through the device in that time.
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