Question

How to solve for t? `3e^t=5+8e^(-t)`

Answer 1

Solution: `t ~~ 0.98`

Explanation:

`3 e^t=5+8 e^(-t)` Multiplying by `e^t` on both sides we get,

`3 e^(2 t)=5 e^t+8` or

`3 e^(2 t) - 5 e^t- 8 = 0` or

`3 e^(2 t) + 3 e^t -8 e^t - 8 = 0` or

`3 e^ t( e^t +1) - 8( e^t + 1) = 0` or

`( e^t +1)(3 e^t - 8) = 0`

`:. e^t = -1 or 3 e^t =8 ; e^t` cannot be negative number

`:. e^t =8/3`, taking natural log on both sides we get,

`t ln e = ln 8 - ln 3 :. t = ln 8 - ln 3 ~~ 0.98 (2 dp)[ ln e=1]`

Solution: `t ~~ 0.98` [Ans]

Answer 2

`t=ln(8/3)~~0.9808`

Explanation:

Here,

`3e^t=5+8e^-t`

`=>3e^t=5+8/(e^t)`

`=>3(e^t)^2=5e^t+8`

`=>3(e^t)^2-5e^t-8=0`

`=>3(e^t)^2+3e^t-8e^t-8=0`

`=>3e^t(e^t+1)-8(e^t+1)=0`

`=>(e^t+1)(3e^t-8)=0`

`=>e^t+1=0 or 3e^t-8=0`

`=>e^t=-1 or e^t=8/3`

`(i)e^t=-1 <0=>e^t !inR^+`

`(ii)e^t=8/3=>t=ln(8/3)~~0.9808`