Q:

Elija got a job as an office clerk in 2000. At the end of every year, Elija has gotten either a raise, a promotion, or both. Elija's salary, S(t), can be determined by multiplying his starting annual salary, $24,400, by the natural logarithm of the product of 3.3 and the number of years since 2000, t. Which of the following correctly models the situation and gives the average rate of change of Elija's salary between 2002 and 2005?

Accepted Solution

A:
Elija's salary: S(t)=$24,400 ln (3.3t)
Number of years since 2000: t

Average rate of change of Elija's salary between 2002 and 2005: r=?
Year 2002β†’t=2002-2000β†’t=2 years
Year 2005β†’t=2005-2000β†’t=5 years

r=[S(b)-S(a)] / (b-a); a=2, b=5
r=[S(5)-S(2)] / (5 years-2 years)
r=[S(5)-S(2)] / (3 years)

t=5β†’S(5)=$24,400 ln [3.3(5)]
S(5)=$24,400 ln(16.5)

t=2β†’S(2)=$24,400 ln [3.3(2)]
S(2)=$24,400 ln(6.6)

r=[S(5)-S(2)] / (3 years)
r=[$24,400 ln(16.5) - $24,400 ln(6.6)] / (3 years)
Common factor $24,400
r=$24,400 [ln(16.5)-ln(6.6)] / (3 years)
r=$24,400 ln(16.5/6.6) / (3 years)
r=$24,400 ln(2.5) / (3 years)
r=$24,400 ln(2.5)/(3 years)
r=$24,400(0.916290732)/(3 years)
r=$22,357.49385/(3 years)
r=$7,452.497952/year
r=$7,452.50/year

The average rate of change of Elija's salary between 2002 and 2005 isΒ $24,400 ln(2.5)/(3 years) or $7,452.50/year