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 term inside the parenthesis: 1 + " , " f r a c 0.055 12 .
Calculate the exponent: 12" , " c d o t 20 = 240 .
Calculate the value of the expression: A = 200" , " t im es " , " f r a c [ ( 1 + 12 0.055 ) ( 12 ⋅ 20 ) − 1 ] ( 12 0.055 ) .
Round the result to the nearest cent: 87125.48 .
Explanation
Understanding the Formula We are given the formula for calculating the future value of an annuity: A = 200" , " t im es " , " f r a c [ ( 1 + 12 0.055 ) ( 12 ⋅ 20 ) − 1 ] ( 12 0.055 ) Our goal is to compute the value of A and round it to the nearest cent.
Calculating the Value First, we need to calculate the value inside the innermost parentheses: 1 + 12 0.055 = 1 + 0.004583333... ≈ 1.004583 Next, we calculate the exponent: 12 ⋅ 20 = 240 Now, we raise the result from the first step to the power calculated in the second step: ( 1.004583 ) 240 ≈ 3.029947 Subtract 1 from the result: 3.029947 − 1 = 2.029947 Divide 0.055 by 12: 12 0.055 ≈ 0.004583 Divide the result from the subtraction by the result from the division: 0.004583 2.029947 ≈ 442.897 Finally, multiply this by 200: 200 × 442.897 ≈ 88579.4 Rounding to the nearest cent, we get $88579.40.
Final Calculation and Rounding Using a calculator, we find that A = 200" , " t im es ( 12 0.055 ) [ ( 1 + 12 0.055 ) ( 12 ⋅ 20 ) − 1 ] ≈ 87125.48 Rounding to the nearest cent, we have $A = 87125.48.
Final Answer Therefore, the value of A rounded to the nearest cent is 87125.48 .
Examples
Understanding compound interest is crucial for financial planning. For instance, if you invest $200 monthly into a retirement account with a 5.5% annual interest rate compounded monthly over 20 years, this calculation helps determine the future value of your investment. Knowing this, you can estimate how much money you'll have saved for retirement, aiding in setting financial goals and making informed decisions about your savings strategy. This calculation provides a clear picture of the long-term growth potential of consistent investments.
The number of electrons flowing through an electric device that delivers a current of 15.0 A for 30 seco n d s is approximately 2.81 × 1 0 21 . This is calculated by determining the total charge delivered and then dividing that by the charge of a single electron. Therefore, about 2.81 sextillion electrons pass through the device.
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