Write an equation to model each situation. In each case, describe how you can tell that it is an example of exponential decay.
a) The value of my car was $25000 when I purchased it. The car depreciates at a rate of 25% each year.
b) The radioactive isotope U238 as a half-life of 4.5x10^9 years.
c) Light intensity in a pond reduced by 12% per meter of depth, relative to the light intensity at the surface.
The formula for compounded growth is Pn = Po*(1+r)^n. Exponential decay is special case where r is negative and there is continuous compounding which gives `P(t) = Po*e^(-r*t)` , with r being the decay constant and t is any quantity expressed in units that is the inverse of the units in which r is expressed.
a) Here, the required equation is: `Pt = Po*e^(-0.25*t)` , where t is in years
b) The half life is 4.5*10^9 years. This gives the decay constant `lambda = ln 2/(4.5*10^9)`
The quantity of U-238 left after t years is `e^(-(ln2/(4.5*10^9))*t)` times the initial quantity or `Q(t) = Qo*e^-(ln2/(4.5*10^9)*t)` .
c) The intensity of light decreases at a constant rate of 12%. `(dI)/(dl) = -0.12` , `I(l) = Io*e^(-0.12*l)` , where l is in meters.
In all the cases it is seen that the rate of decay is proportional to the time or length. This shows that the decrease is exponential in nature.