An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (2)
The total charge flowing through a device with a current of 15.0 A for 30 seconds is 450 coulombs. This charge corresponds to approximately 2.81 billion billion electrons flowing through the device. Therefore, approximately 2.81 × 1 0 21 electrons pass through the device in that time frame.
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Set up the compound interest formula: 3500 ( 1.08 ) t < 6000 .
Simplify the inequality: ( 1.08 ) t < 7 12 .
Apply logarithms: t ⋅ l n ( 1.08 ) < l n ( 7 12 ) .
Solve for t : t < l n ( 1.08 ) l n ( 7 12 ) ≈ 7.003 , so t < 7 .
The number of years is t < 7 .
Explanation
Problem Analysis Let's analyze the problem. We are given an initial investment of $3,500 with an annual interest rate of 8%. We want to find the number of years, t , for which the account balance will be less than $6,000.
Compound Interest Formula The formula for compound interest is: A = P ( 1 + r ) t where:
A is the amount after t years
P is the principal amount (initial investment)
r is the annual interest rate
t is the number of years
Setting up the Inequality In this case, P = 3500 , r = 0.08 , and we want to find t such that A < 6000 . So, we have: 3500 ( 1 + 0.08 ) t < 6000
Simplifying the Inequality 3500 ( 1.08 ) t < 6000
Divide both sides by 3500: ( 1.08 ) t < 3500 6000 ( 1.08 ) t < 7 12
Applying Logarithms To solve for t , we can take the natural logarithm (ln) of both sides: l n (( 1.08 ) t ) < l n ( 7 12 ) Using the property of logarithms, l n ( a b ) = b ⋅ l n ( a ) :
t ⋅ l n ( 1.08 ) < l n ( 7 12 )
Isolating t Now, divide both sides by l n ( 1.08 ) :
t < l n ( 1.08 ) l n ( 7 12 )
Calculating the Value Using a calculator, we find: t < l n ( 1.08 ) l n ( 12/7 ) ≈ 0.0770 0.5390 ≈ 7.003
Determining the Number of Years Since t must be less than approximately 7.003, the largest whole number of years for which the account will have less than $6,000 is 7. Therefore, t < 7 .
Final Answer The number of years for which the account will have less than $6,000 is t < 7 .
Examples
Understanding compound interest is crucial for making informed financial decisions. For instance, when planning for retirement, individuals need to estimate how their investments will grow over time. By using the compound interest formula, they can project the potential value of their savings and make adjustments to their investment strategies as needed. This helps ensure they reach their financial goals by the time they retire. The formula A = P ( 1 + r ) t allows you to calculate the future value ( A ) of an investment based on the initial principal ( P ), the annual interest rate ( r ), and the number of years ( t ).
