Respuesta :
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: [tex]y=a*b^x[/tex], 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.
[tex]22.6\%=\frac{22.6}{100}=0.226[/tex]
Upon substituting our given values in exponential decay function we will get,
[tex]y=650*(1-0.226)^x[/tex]
[tex]y=650*(0.774)^x[/tex]
Therefore, the function [tex]y=650*(0.774)^x[/tex] 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.
[tex]40=650*(0.774)^x[/tex]
Let us divide both sides of our equation by 650.
[tex]\frac{40}{650}=\frac{650*(0.774)^x}{650}[/tex]
[tex]\frac{4}{65}=(0.774)^x[/tex]
Let us take natural log of both sides of our equation.
[tex]ln(\frac{4}{65})=ln(0.774)^x)[/tex]
Using logarithm property [tex]ln(a^x)=x*ln(a)[/tex] we will get,
[tex]ln(\frac{4}{65})=x*ln(0.774)[/tex]
[tex]-2.7880929087757464=x*-0.2561834053924099[/tex]
[tex]x=\frac{-2.7880929087757464}{-0.2561834053924099}[/tex]
[tex]x=10.88319\approx 11[/tex]
Therefore, it will take 11 minutes to until there are 40 grams of the element remaining.
