19.03.2023 - 15:27

Your starting salary at a company is 65000 dollars, and you get a raise of 4000 dollars each year. Let t represent the number of years since you joined the company. a) Express the percentage rate of change of your salary as a function of time. b) At w

Your starting salary at a company is 65000 dollars, and you get a raise of 4000 dollars each year. Let t represent the number of years since you joined the company.

a) Express the percentage rate of change of your salary as a function of time.

b) At what percentage rate will your salary be increasing after 2 years?

c) What value does the percentage rate of change of your salary approach in the long run?

Answers (1)
  • antoninho
    April 2, 2023 в 03:17

    a) The percentage rate of change of your salary as a function of time can be expressed as follows:

    Let S(t) be your salary t years after joining the company.

    Then, S(t) = 65000 + 4000t (since you get a raise of $4000 each year).

    The percentage rate of change of your salary is given by:

    [(S(t+1) - S(t)) / S(t)] * 100

    Substituting the expression for S(t) into the above formula, we get:

    [(65000 + 4000(t+1) - (65000 + 4000t)) / (65000 + 4000t)] * 100

    Simplifying the above expression, we get:

    [(4000 / (65000 + 4000t))] * 100

    Therefore, the percentage rate of change of your salary as a function of time is:

    f(t) = (4000 / (65000 + 4000t)) * 100

    b) After 2 years, the percentage rate of change of your salary can be found by substituting t = 2 into the expression for f(t):

    f(2) = (4000 / (65000 + 4000*2)) * 100

    Simplifying the above expression, we get:

    f(2) = 5.80%

    Therefore, your salary will be increasing at a rate of 5.80% after 2 years.

    c) In the long run, as t approaches infinity, the denominator of the expression for f(t) becomes very large compared to the numerator. Therefore, the percentage rate of change of your salary approaches zero. In other words, your salary will continue to increase each year, but the rate of increase will become smaller and smaller as time goes on. This is because the fixed $4000 increase becomes a smaller and smaller percentage of your overall salary as your salary grows larger.

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