Question

An element with the mass of 650 grams decay by 22.6% per minute. to the nearest minute,how long will it be until there are 40 grams of the element remaining

Answer 1

11 minutes.

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:
`y=a*b^x`, 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\%=(22.6)/(100)=0.226`

Upon substituting our given values in exponential decay function we will get,

`y=650*(1-0.226)^x`

`y=650*(0.774)^x`

Therefore, the function
`y=650*(0.774)^x` 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`

Let us divide both sides of our equation by 650.

`(40)/(650)=(650*(0.774)^x)/(650)`

`(4)/(65)=(0.774)^x`

Let us take natural log of both sides of our equation.

`ln((4)/(65))=ln(0.774)^x)`

Using logarithm property
`ln(a^x)=x*ln(a)` we will get,

`ln((4)/(65))=x*ln(0.774)`

`-2.7880929087757464=x*-0.2561834053924099`

`x=(-2.7880929087757464)/(-0.2561834053924099)`

`x=10.88319\approx 11`

Therefore, it will take 11 minutes to until there are 40 grams of the element remaining.