Answer:
11 minutes.
Step-by-step explanation:
Let x be the number of minutes.
We have been given that an element with the mass of 650 grams decay by 22.6% per minute.
As mass of the element is decaying 22.6% per minute, so this situation can be modeled by exponential function.
Since an exponential function is in form:
, where,
a = Initial value,
b = For decay b is in form (1-r), where r is decay rate in decimal form.
Let us convert our given decay rate in decimal form.
![22.6\%=\frac{22.6}{100}=0.226](https://tex.z-dn.net/?f=22.6%5C%25%3D%5Cfrac%7B22.6%7D%7B100%7D%3D0.226)
Upon substituting our given values in exponential decay function we will get,
![y=650*(0.774)^x](https://tex.z-dn.net/?f=y%3D650%2A%280.774%29%5Ex)
Therefore, the function
represents the remaining mass of element (y) after x minutes.
To find the number of minutes it will take until there are 40 grams of the element remaining we will substitute y=40 in our function.
![40=650*(0.774)^x](https://tex.z-dn.net/?f=40%3D650%2A%280.774%29%5Ex)
Let us divide both sides of our equation by 650.
![\frac{40}{650}=\frac{650*(0.774)^x}{650}](https://tex.z-dn.net/?f=%5Cfrac%7B40%7D%7B650%7D%3D%5Cfrac%7B650%2A%280.774%29%5Ex%7D%7B650%7D)
![\frac{4}{65}=(0.774)^x](https://tex.z-dn.net/?f=%5Cfrac%7B4%7D%7B65%7D%3D%280.774%29%5Ex)
Let us take natural log of both sides of our equation.
![ln(\frac{4}{65})=ln(0.774)^x)](https://tex.z-dn.net/?f=ln%28%5Cfrac%7B4%7D%7B65%7D%29%3Dln%280.774%29%5Ex%29)
Using logarithm property
we will get,
![ln(\frac{4}{65})=x*ln(0.774)](https://tex.z-dn.net/?f=ln%28%5Cfrac%7B4%7D%7B65%7D%29%3Dx%2Aln%280.774%29)
![-2.7880929087757464=x*-0.2561834053924099](https://tex.z-dn.net/?f=-2.7880929087757464%3Dx%2A-0.2561834053924099)
![x=\frac{-2.7880929087757464}{-0.2561834053924099}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B-2.7880929087757464%7D%7B-0.2561834053924099%7D)
![x=10.88319\approx 11](https://tex.z-dn.net/?f=x%3D10.88319%5Capprox%2011)
Therefore, it will take 11 minutes to until there are 40 grams of the element remaining.