Math Help

Growth and Decay Applications

Example: An initial sample contains 100 cells and grows at a constant rate. After 8 hours

there are 2000 cells. Determine the growth rate, find an expression, y(t), for the population

of cells at any time t , and determine the number of cells after 24 hours.

Strategy: Apply

y(t) = C e ^kt

where y(t) is the size of the population at any time t

y(0) = 100 = C = initial population

k is the growth rate (to be found)

y(8) = 100 e ^8k = 2000

e ^8k = 20 So 8k = ln(20) and k = (1/8) ln(20)

k = ln ( 20 ^1/8 ) (result)

So y(t) = 100 exp[(ln [ 20 ^1/8 ) t ]) = (100) 20^t/8 (result)

y(24) = (100) 20³ = 800,000 cells (result)

Example: A sample of 50 cancer cells are treated with radiation such that half of them

are killed in 28 days. Assume a constant decay rate. Find the decay rate, the general

expression for the number of cancer cells after t days, and how many cells still survive

after 40 days.

Strategy: Apply y(t) = C e ^kt y(0) = 50

y(t) = 50 e ^kt

25/50 = 1/2 = e ^28k

28 k = ln (1/2), k = (1/28) ln (1/2) = ˗ ln (2^1/28) (result)

y(t) = 50 exp [˗ ln (2^1/28) t ] ] = (50) 2^{˗ t/28} (result)

y(40) = (50) 2^{˗ 40/28} = (50) 2^{˗ 10/7} ≈ 19 cells (result)