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...
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.