Question

The function A = A 0 e -0.0077x models the amount in pounds of a particular radioactive material stored in a concrete vault, where x is the number of years since the material was put into the vault. If 800 pounds of the material are placed in the vault, how much time will need to pass for only 432 pounds to remain?

Answer 1

After 80 years 432 pounds of radioactive element will be remaining.

Explanation:

A function
`A_(t)=A_(0)e^(-0.0077x)` models the amount of a radioactive material.

Here x = number of years taken for decay.

If
`A_(t)=432` pound and
`A_(0)=800` pound, then we have to calculate the time for decay.

`A_(t)=A_(0)e^(-0.0077x)`

`432=800e^(-0.0077x)`

`e^(-0.0077x)=(432)/(800)`

Take log on both the sides of the equation

`ln(e^(-0.0077x))=ln((432)/(800) )`

-0.0077x = ln(0.54)

-0.0077x = -0.616186

x =
`(0.616186)/(0.0077)`

x = 80.02 years

Therefore, After 80 years 432 pounds of radioactive element will be remaining.