SOLUTION: the function A=Aoe-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 in

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: the function A=Aoe-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 in      Log On


   

Question 239904: the function A=Aoe-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 700 punds of the material are placed in the vault, how much time will need to pass for only 150 pounds to remain?
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
If the equation is A+=+A%5B0%5De%5E%28-0.0077x%29, then read on. If not then repost your question with a correct and clear equation, using parentheses generously.

When a subscript of 0 is used, it is often used to indicate the initial or starting value for that value. In this case, A%5B0%5D represents the initial amount of the radioactive material. So we can rephrase the problem as: If A%5B0%5D+=+700, after how many years will the amount, A, be 150? So we want to solve the equation:
150+=+700e%5E%28-0.0077x%29
To solve for variables in the exponent like this, we often use logarithms. But first we will isolate the e and its exponent by dividing both sides by 700:
150%2F700+=+e%5E%28-0.0077x%29
The fraction on the left will reduce to:
3%2F14+=+e%5E%28-0.0077x%29
Now we will use logarithms. We can use any logarithm which our calculator can handle. The "natural" choice (excuse the pun) is to use natural logarithms because of the e. But in case your calculator only does base 10 logarithms I will finish the solution using both:
  • Natural logarithms.
    • Find the natural logarithm of each side:
      ln%283%2F14%29+=+ln%28e%5E%28-0.0077x%29%29
    • Use the property log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29 to move the exponent out in front:
      ln%283%2F14%29+=+%28-0.0077x%29ln%28e%29
    • Since ln(e) = 1:
      ln%283%2F14%29+=+-0.0077x
    • Divide both sides by -0.0077:
      ln%283%2F14%29%2F%28-0.0077%29+=+x
    • Use your calculator to simplify the left side
      %28-1.5404450409471489%29%2F%28-0.0077%29+=+x
      200.0577975256037562+=+x
  • Base 10 (aka Common) logarithms.
    • Find the base 10 logarithm of each side:
      log%28%283%2F14%29%29+=+log%28%28e%5E%28-0.0077x%29%29%29
    • Use the property log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29 to move the exponent out in front:
      log%28%283%2F14%29%29+=+%28-0.0077x%29log%28%28e%29%29
    • Divide both sides by %28-0.0077%2Alog%28%28e%29%29%29:
      log%28%283%2F14%29%29%2F%28-0.0077%2Alog%28%28e%29%29%29+=+x
    • Substitute the approximation 2.7182818284590451 for e. (Feel free to round this off some.):
      log%28%283%2F14%29%29%2F%28-0.0077%2Alog%28%282.7182818284590451%29%29%29+=+x

    • Use your calculator to simplify the left side
      %28-0.6690067809585756%29%2F%28-0.0077%2A0.4342944819032518%29+=+x
      %28-0.6690067809585756%29%2F%28-0.0033440675106550%29+=+x
      200.0577975256037668+=+x

So it will be about 200 years before the 700 pounds will be reduced to 150 pounds. (Note that we get the same answer using either natural or base 10 logarithms. (The very slight difference is due to using an approximation for e.))