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Find the present value of the income \(c\) (measured in dollars) over \(t_{1}\) years at the given annual inflation rate \(r\). $$ c=30,000+500 t, r=7 \%, t_{1}=6 \text { years } $$

Find the present value of the income \(c\) (measured in dollars) over \(t_{1}\) years at the given annual inflation rate \(r\). $$ c=1000+50 e^{t / 2}, r=6 \%, t_{1}=4 \text { years } $$

Find the present value of the income \(c\) (measured in dollars) over \(t_{1}\) years at the given annual inflation rate \(r\). $$ c=5000+25 t e^{t / 10}, r=6 \%, t_{1}=10 \text { years } $$

Present Value A company expects its income \(c\) during the next 4 years to be modeled by \(c=150,000+75,000 t\) (a) Find the actual income for the business over the 4 years. (b) Assuming an annual inflation rate of \(4 \%\), what is the present value of this income?

Present Value A professional athlete signs a three-year contract in which the earnings can be modeled by \(c=300,000+125,000 t .\) (a) Find the actual value of the athlete's contract. (b) Assuming an annual inflation rate of \(3 \%\), what is the present value of the contract?

Find the future value of the income (in dollars) given by \(f(t)\) over \(t_{1}\) years at the annual interest rate of \(r\). If the function \(f\) represents a continuous investment over a period of \(t_{1}\) years at an annual interest rate of \(r\) (compounded continuously), then the future value of the investment is given by Future value \(=\mathrm{e}^{r t_{1}}\left[{ }^{t_{1}} f(t) e^{-r t} d t\right.\). $$ f(t)=3000, r=8 \%, t_{1}=10 \text { years } $$

Find the future value of the income (in dollars) given by \(f(t)\) over \(t_{1}\) years at the annual interest rate of \(r\). If the function \(f\) represents a continuous investment over a period of \(t_{1}\) years at an annual interest rate of \(r\) (compounded continuously), then the future value of the investment is given by Future value \(=\mathrm{e}^{r t_{1}}\left[{ }^{t_{1}} f(t) e^{-r t} d t\right.\). $$ f(t)=3000 e^{0.05 t}, r=10 \%, t_{1}=5 \text { years } $$

Finance: Future Value Use the equation from Exercises 77 and 78 to calculate the following. (Source: Adapted from Garman/Forgue, Personal Finance, Eighth Edition) (a) The future value of \(\$ 1200\) saved each year for 10 years earning \(7 \%\) interest. (b) A person who wishes to invest \(\$ 1200\) each year finds one investment choice that is expected to pay \(9 \%\) interest per year and another, riskier choice that may pay \(10 \%\) interest per year. What is the difference in return (future value) if the investment is made for 15 vears?

Use a program similar to the Midpoint Rule program on page 856 with \(n=12\) to approximate the area of the region bounded by the graphs of \(y=\frac{10}{\sqrt{x} e^{x}}, y=0, x=1\), and \(x=4\).