An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (1)
Calculate Claire's salary in 2001 and 2004 using the function C ( t ) = 40 , 000 + 1 , 000 ln ( t + 1 ) .
Calculate the average rate of change for Claire: 4 − 1 C ( 4 ) − C ( 1 ) ≈ 305.43 .
Determine Declan's salary in 2001 and 2004 from the table and calculate the average rate of change for Declan: 4 − 1 D ( 4 ) − D ( 1 ) ≈ 274.89 .
Compare the average rates of change and conclude that Claire had a greater average rate of change: Cl ai re .
Explanation
Calculate Claire's Salaries First, we need to calculate Claire's salary in 2001 ( t = 1 ) and 2004 ( t = 4 ) using the given function: C ( t ) = 40 , 000 + 1 , 000 ln ( t + 1 ) .
Calculate Salaries for 2001 and 2004 Claire's salary in 2001: C ( 1 ) = 40 , 000 + 1 , 000 ln ( 1 + 1 ) = 40 , 000 + 1 , 000 ln ( 2 ) ≈ 40 , 000 + 1 , 000 ( 0.6931 ) = 40 , 693.15 Claire's salary in 2004: C ( 4 ) = 40 , 000 + 1 , 000 ln ( 4 + 1 ) = 40 , 000 + 1 , 000 ln ( 5 ) ≈ 40 , 000 + 1 , 000 ( 1.6094 ) = 41 , 609.44
Calculate Claire's Average Rate of Change Next, we calculate the average rate of change in Claire's salary between 2001 and 2004: 4 − 1 C ( 4 ) − C ( 1 ) = 3 41 , 609.44 − 40 , 693.15 = 3 916.29 ≈ 305.43 So, Claire's average rate of change is approximately $305.43.
Find Declan's Salaries from the Table Now, we determine Declan's salary in 2001 and 2004 from the table: Declan's salary in 2001: $D(1) = 45 , 623.83 Dec l a n ′ ss a l a ry in 2004 : D(4) = $46,448.49
Calculate Declan's Average Rate of Change We calculate the average rate of change in Declan's salary between 2001 and 2004: 4 − 1 D ( 4 ) − D ( 1 ) = 3 46 , 448.49 − 45 , 623.83 = 3 824.66 ≈ 274.89 So, Declan's average rate of change is approximately $274.89.
Compare and Conclude Finally, we compare the average rates of change for Claire and Declan: Claire's average rate of change: $305.43 Declan's average rate of change: $274.89 Since 274.89"> 305.43 > 274.89 , Claire had a greater average rate of change in pay between 2001 and 2004.
Final Answer Therefore, Claire had the greater average rate of change in pay between 2001 and 2004.
Examples
Understanding average rates of change can help you make informed financial decisions. For example, if you're comparing two investment options, calculating the average rate of return over a period of time can help you determine which investment is likely to yield better results. This concept is also useful in project management, where you might track the average rate of progress to ensure a project stays on schedule. By comparing these rates, you can make strategic adjustments to maximize your returns or improve project efficiency.
