(126) times (1/2 to the x/11 power) = 15% times 126
(1/2) to the x/11 power = 0.15
Take the log of both sides :
(x/11) times log(1/2) = log(0.15)
Multiply both sides by 11 :
'x' times log(1/2) = 11 x log(0.15)
Divide both sides by " log(1/2) " :
x = 11 x log(0.15)/log(1/2) = <em><u>30.107 years</u></em> (rounded)
That's the time it takes for <em><u>any-size</u></em> sample of this substance
to decay to 15% of its original size.
Answer:

Step-by-step explanation:
We are given that:

And we want to find F'(0).
First, find F(x):
![\displaystyle F'(x) = \frac{d}{dx}\left[ f(3x)]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20F%27%28x%29%20%3D%20%5Cfrac%7Bd%7D%7Bdx%7D%5Cleft%5B%20f%283x%29%5D)
From the chain rule:
![\displaystyle \begin{aligned} F'(x) &= f'(3x) \cdot \frac{d}{dx} \left[ 3x\right] \\ \\ &= 3f'(3x)\end{aligned}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cbegin%7Baligned%7D%20F%27%28x%29%20%26%3D%20f%27%283x%29%20%5Ccdot%20%5Cfrac%7Bd%7D%7Bdx%7D%20%5Cleft%5B%203x%5Cright%5D%20%5C%5C%20%5C%5C%20%26%3D%203f%27%283x%29%5Cend%7Baligned%7D)
Then:

In conclusion, F'(0) = 15.