Quotient rule with additional condition

[Fluidman117]

Homework Statement

I am working on chemical reaction engineering problem and it involves some math, which I am not able to figure out...

I have to find the residence time for maximum production, which is in the case when : (dη_p)/dτ=0

I have to find the τ (residence time).

Homework Equations

η_p=0.1τ/(1+0.4τ+0.04τ^2)

The Attempt at a Solution

I figure I have to use the quotient rule:

d[f/g]=(f'g-fg')/g^2

f=0.1τ
g=1+0.4τ+0.04τ^2

When I use the rule, I come up with an number which is not the same as in the answers. The reason is I think that I don't know how to take into account the condition that (dη_p)/dτ=0

 

[ehild]

The derivative has to be zero, that means an equation for τ, easy to solve. Show your further work please.

ehild

 

[Fluidman117]

When you say the derivative equals zero, do you mean this:

0=(f'g-fg')/g^2

0=[0.1(1+0.4τ+0.04τ^2)-(0.1τ)(0.4+0.08τ)]/g^2
0=0.1-0.004τ^2/g^2

0.1-0.004τ^2=0

and solving for τ=5 , which is the correct answer.

However, I don't understand why g^2=0 ?

 

[HallsofIvy]

g^2 is NOT 0. What makes you think it is?

 

[Fluidman117]

I don't have any scientific reasoning behind my logic, but basically that is the only way I could see that I come up with the answer given in the book.If I subsitute g^2=(1+0.4τ+0.04τ^2)^2 and using the formula
(a+b+c)^2= a^2+b^2+c^2+ 2(ab+bc+ca)

I end up with

0=(f'g-fg')/g^2=(0.1-0.004τ^2)/(1+0.8τ+0.24τ^2+0.032τ^3+0.0016τ^4)

What would be the next step from here?

 

[SteamKing]

When you say the derivative equals zero, do you mean this:

0=(f'g-fg')/g^2

0=[0.1(1+0.4τ+0.04τ^2)-(0.1τ)(0.4+0.08τ)]/g^2
0=0.1-0.004τ^2/g^2

0.1-0.004τ^2=0

and solving for τ=5 , which is the correct answer.

However, I don't understand why g^2=0 ?

If g^2 = 0, the derivative is undefined. The quantity (f'g - fg') must be 0 to satisfy the equation.

 

[Mark44]

If g^2 = 0, the derivative is undefined. The quantity (f'g - fg') must be 0 to satisfy the equation.

To elaborate on what SteamKing said, the basic idea is that if a/b = 0, then a = 0. Of course, b cannot be 0.

 

[Fluidman117]

Thanks everyone!