Assume an exponential function has a starting value of 19 and a decay rate of 9%. Write an equation to model the situation.

Question

Type an expression using x a variable. Use integers or decimals for any numbers in the expression.

Answer 1

f(x) = ab^x, a is initial, b is rate, x is time

Note a decay of 0.09 means that 0.91 remains.

f(x) = 19(0.91)^x

Answer 2

A=Pe^rt is a general formula for continuous growth or decay

if r=0, there is no change, no growth or decay and the formula collapses to A=Pe^(0)t = P

if r>0, it's growth

if r<0 it's decay

it's an exponential function where the variable is the exponent, with a constant base and constant coefficient

P=19, t or x =number of years, r=the growth or decay rate= 9% = .09 per year

A = the amount left after P decays for x number of years

A=19e^-.09x

or

A=19(1+r/n)^nt where n= number of compounding periods per year

if n=1

then

A=19(1+r)^t

if r=-.09, a decay rate of 9%, then

A=19(1-.09)^t

A=19(.91)^t

or

A = 19(.91)^x

try one year,

x=1

A=19(.91)^1 = 19(.91) = 17.29

or the continuous compounding decay formula

which is the same as in contuously compounded interest in financial economics problems or present value problems

A=19e^-.09(1) = 19e^-.09 =19/e^.09 = about 17.36

there is a slight difference for one year

but the discrepancy gets huge for very large time periods

decay in the natural world, whether radioactive chemicals, carbon 14, plants dying, humans aging or disease occurs "naturally" at a continuous decay rate. "natural" shows up in the natural logs whose inverse is the exponential function with base e

to view decay as occurring with a compounding period of once a year is not a realistic model, although it does provide a very close approximation for short time periods