where Q 0 is the initial amount and k is the (negative) factor of proportionality. Since it is given
that when t = 1612, equation (1) yieldsWe now haveWhen one quarter of Q 0 has decayed, three quarters of the initial amount remains. We use this
fact in equation (2) to find t:Applications of Exponential Growth
(1) A colony of bacteria may grow at a rate proportional to its size.
(2) Other populations, such as those of humans, rodents, or fruit flies, whose supply of food is
unlimited may also grow at a rate proportional to the size of the population.
(3) Money invested at interest that is compounded continuously accumulates at a rate proportional to
the amount present. The constant of proportionality is the interest rate.
(4) The demand for certain precious commodities (gas, oil, electricity, valuable metals) has been
growing in recent decades at a rate proportional to the existing demand.
Each of the above quantities (population, amount, demand) is a function of the form cekt (with k >
0). (See Figure N9–7a.)
(5) Radioactive isotopes, such as uranium-235, strontium-90, iodine-131, and carbon-14, decay at a
rate proportional to the amount still present.
(6) If P is the present value of a fixed sum of money A due t years from now, where the interest is
compounded continuously, then P decreases at a rate proportional to the value of the investment.
(7) It is common for the concentration of a drug in the bloodstream to drop at a rate proportional to
the existing concentration.
(8) As a beam of light passes through murky water or air, its intensity at any depth (or distance)
decreases at a rate proportional to the intensity at that depth.