Consider an income stream of R(t) dollars per year, earning interest at the rate of r per year compounded continuously. The present value of the income stream R(t) over a term of T years is: PV = R(t)e dt. When the interval of time is infinite, we refer to the present value as the capital value of the income stream. Notice that the integrand is a form of the continuous exponential model, which can be used to model any type of continuous growth or decay (depending on the sign of the exponent), and is thus regarded as a common and widely applicable model. (1) A company is considering expanding their production capabilities with a new machine that costs $95,000 and has a projected lifespan of 8 years. They estimate the increased production will provide a constant $12,500 per year of additional income. The income stream can earn 0.6% per year, compounded continuously. Should the company buy the machine? == (2) Suppose that an organization wants to establish a trust fund that will provide a contin- uous income stream with an annual rate of flow at time t given by R(t) = 10,000. If the interest rate remains at 8%, compounded continuously, find the capital value of the fund.