Question

The function a = ape- 001386x 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. Suppose 400 pounds of the material are initially put into the vault. After how many years will there be 50 pounds left?

Answer 1

To find the number of years it will take for 50 pounds of the radioactive material to remain in the vault, we can use the given function a = ape- 0.01386x. By substituting the initial amount of 400 pounds and setting the equation equal to 50 pounds, we can solve for x to find that the 50 pounds will be left after 0 years.

Step-by-step explanation:

To find the number of years it will take for 50 pounds of the radioactive material to remain in the vault, we need to set up an equation using the given function. The given function is a = ape- 0.01386x, where a represents the amount of material in pounds and x represents the number of years since the material was put into the vault. We are given that the initial amount of material is 400 pounds, so we can substitute a = 400 into the equation:

400 = 400e¯0.01386x

Next, we can divide both sides of the equation by 400 to simplify:

e¯0.01386x = 1

Since e¯0.01386x is equal to 1, the exponent must be 0. Therefore, we can solve for x:

0 = 0.01386x

Dividing both sides by 0.01386, we get:

x = 0

So after 0 years, meaning immediately, there will be 50 pounds left in the vault.