after graduating from college, carlos receives two different job offers. both pay a starting salary of…
after graduating from college, carlos receives two different job offers. both pay a starting salary of $61000 per year, but one job promises a $3660 raise per year, while the other guarantees a 5% raise each year. complete the tables below to determine what his salary will be after t years. round your answers to the nearest dollar. t years 1 5 10 15 20 salary with $3660 raise per year t years 1 5 10 15 20 salary with 5% raise per year
Answer
Explanation:
Step1: Find the formula for the first - type of raise
The starting salary $a = 61000$ and the annual raise is $d=3660$. The salary $S_1$ after $t$ years is given by the arithmetic - sequence formula $S_1=a + dt$.
Step2: Calculate $S_1$ for $t = 1$
Substitute $a = 61000$, $d = 3660$, and $t = 1$ into the formula $S_1=a+dt$. So $S_1=61000+3660\times1=64660$.
Step3: Calculate $S_1$ for $t = 5$
Substitute $a = 61000$, $d = 3660$, and $t = 5$ into the formula $S_1=a+dt$. So $S_1=61000+3660\times5=61000 + 18300=79300$.
Step4: Calculate $S_1$ for $t = 10$
Substitute $a = 61000$, $d = 3660$, and $t = 10$ into the formula $S_1=a+dt$. So $S_1=61000+3660\times10=61000 + 36600=97600$.
Step5: Calculate $S_1$ for $t = 15$
Substitute $a = 61000$, $d = 3660$, and $t = 15$ into the formula $S_1=a+dt$. So $S_1=61000+3660\times15=61000+54900 = 115900$.
Step6: Calculate $S_1$ for $t = 20$
Substitute $a = 61000$, $d = 3660$, and $t = 20$ into the formula $S_1=a+dt$. So $S_1=61000+3660\times20=61000 + 73200=134200$.
Step7: Find the formula for the second - type of raise
The starting salary $P = 61000$ and the growth rate $r=0.05$. The salary $S_2$ after $t$ years is given by the compound - interest formula $S_2=P(1 + r)^t$.
Step8: Calculate $S_2$ for $t = 1$
Substitute $P = 61000$, $r = 0.05$, and $t = 1$ into the formula $S_2=P(1 + r)^t$. So $S_2=61000\times(1 + 0.05)^1=61000\times1.05=64050$.
Step9: Calculate $S_2$ for $t = 5$
Substitute $P = 61000$, $r = 0.05$, and $t = 5$ into the formula $S_2=P(1 + r)^t$. So $S_2=61000\times(1 + 0.05)^5=61000\times1.05^5\approx61000\times1.27628=77853$.
Step10: Calculate $S_2$ for $t = 10$
Substitute $P = 61000$, $r = 0.05$, and $t = 10$ into the formula $S_2=P(1 + r)^t$. So $S_2=61000\times(1 + 0.05)^{10}=61000\times1.05^{10}\approx61000\times1.62889=99362$.
Step11: Calculate $S_2$ for $t = 15$
Substitute $P = 61000$, $r = 0.05$, and $t = 15$ into the formula $S_2=P(1 + r)^t$. So $S_2=61000\times(1 + 0.05)^{15}=61000\times1.05^{15}\approx61000\times2.07893=126815$.
Step12: Calculate $S_2$ for $t = 20$
Substitute $P = 61000$, $r = 0.05$, and $t = 20$ into the formula $S_2=P(1 + r)^t$. So $S_2=61000\times(1 + 0.05)^{20}=61000\times1.05^{20}\approx61000\times2.65329=161850$.
Answer:
| $t$ years | 1 | 5 | 10 | 15 | 20 |
|---|---|---|---|---|---|
| Salary with $3660$ raise per year | $64660$ | $79300$ | $97600$ | $115900$ | $134200$ |
| Salary with 5% raise per year | $64050$ | $77853$ | $99362$ | $126815$ | $161850$ |