Question

Two-thousand ounces of a radioactive substance are stored in a radioactive container. The amount, in ounces, of the substance that is left after t years can be modeled by the equation y=2000e−0.00043t, where y is the amount of the substance left after t years.Approximately how many years will it take until 50 ounces of this substance remains?

Answer 1

Answer:

Approximately 1,933.64 years, it will take until 50 ounce of this substance remains.

Step-by-step explanation:

Given that,

The amount, in ounce, of the substance that is left after t years can be modeled by the equation

$y=2000e^{-0.00043t}$

To find the time, we put y = 50 ounce in the above equation.

$50=2000e^{-0.00043t}$

$\Rightarrow\frac{ 50}{2000}=e^{-0.00043t}$

$\Rightarrow\frac{ 1}{400}=e^{-0.00043t}$

Taking ln function both sides of the above equation

$\Rightarrow ln|\frac{ 1}{400}|=ln|e^{-0.00043t}|$

$\Rightarrow ln|{ 1}|-ln|{400}|=ln|e^{-0.00043t}|$   [ since $ln| \frac ab|= ln|a|- ln| b|$  ]

$\Rightarrow -ln|{400}|={-0.00043t}$               [ since $ln|1|=0$ and $e^{ln |a|}=a$  ]

$\Rightarrow t = \frac{ln|{400}|}{0.00043}$

      ≈1,933.64 years.

Approximately 1,933.64 years, it will take until 50 ounce of this substance remains.