Question

The function A=A0 e-⁰.⁰⁰⁹⁹⁰ˣ 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 488 pounds to remain?

Answer 1

The amount of time needed for only 488 pounds of the radioactive material to remain can be found by solving the equation 488 = 800e^-0.0090x. After solving for x, we find that approximately 257.6 years will need to pass.

Step-by-step explanation:

The function A=A0 e^-0.0090x models the amount in pounds of the radioactive material stored in the 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, we can use this equation to find out how much time will need to pass for only 488 pounds to remain.

Set up the equation:

488 = 800e^-0.0090x

Divide both sides by 800:

0.61 = e^-0.0090x

Take the natural logarithm of both sides:

ln(0.61) = -0.0090x

Divide both sides by -0.0090:

x = ln(0.61) / -0.0090

Use a calculator to evaluate the right side of the equation:

x ≈ 257.6 years

Therefore, approximately 257.6 years will need to pass for only 488 pounds of the radioactive material to remain.