Question

Need help with a growth problem ?

Answer 1

• `p(t) =10^6 (1 - e^(-1/8 ln (4/3)t))`

• `p(20) approx 512,861`

Explanation:

`dot p = k( 10^6 - p)`

`int (d p)/( 10^6 - p) = k int dt`

`-ln( 10^6 - p) = kt + C`

`10^6 - p = Ce^(-kt)`

`p =10^6 - Ce^(-kt)`

Seems reasonable to assume that: `p(0) = 0`

`implies p =10^6 (1 - e^(-kt))`

With the condition:

• `p(8) = 10^6/4`

`implies k = 1/8 ln (4/3)`

So the formula is:

• `p(t) =10^6 (1 - e^(-1/8 ln (4/3)t))`

After 20 weeks:

• `p(20) =10^6 (1 - exp(-1/8 ln (4/3)* 20))`

`approx 512,861`