Constant Doubling Time Functions: Exponential vs. Linear | Homework Help

[physicsernaw]

Homework Statement

The only functions with a constant doubling time are the exponential functions P₀e^kt with k > 0. Show that the doubling time of linear function f(t) = a(t) + b at time t₀ is t₀ + b/a.

Homework Equations

n/a

The Attempt at a Solution

With initial time t₀, P = at₀ + b

At some time t, P is doubled: at + b = 2P

t = (2P - b)/a

Plug in P and simplify:

t = (2at₀ + b)/a
t = 2t₀ + b/a

What am I doing wrong?

 

[Dick]

Homework Statement

The only functions with a constant doubling time are the exponential functions P₀e^kt with k > 0. Show that the doubling time of linear function f(t) = a(t) + b at time t₀ is t₀ + b/a.

Homework Equations

n/a

The Attempt at a Solution

With initial time t₀, P = at₀ + b

At some time t, P is doubled: at + b = 2P

t = (2P - b)/a

Plug in P and simplify:

t = (2at₀ + b)/a
t = 2t₀ + b/a

What am I doing wrong?

You haven't done anything wrong. But I think they mean the doubling time to be t-t0. That's the length of time after t0 that the function will double. Solve for that.

 

[physicsernaw]

You haven't done anything wrong. But I think they mean the doubling time to be t-t0. That's the length of time after t0 that the function will double. Solve for that.

Ahhh, thank you so much.