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

To find approximately how many years it will take until 50 ounces of the radioactive substance remains, we can set y to 50 in the equation y=2000e^(-0.00043t) and solve for t.

Step-by-step explanation:

The given equation, y=2000e^(-0.00043t), models the amount of radioactive substance left after t years.

To find approximately how many years it will take until 50 ounces of the substance remains, we can set y to 50 and solve for t.

First, plug in 50 for y in the equation: 50 = 2000e^(-0.00043t)

Divide both sides by 2000: e^(-0.00043t) = 0.025

Take the natural logarithm of both sides: ln(e^(-0.00043t)) = ln(0.025)

Using the property of logarithms, the exponent can be brought down as a coefficient: -0.00043t = ln(0.025)

Finally, divide both sides by -0.00043 to solve for t: t = ln(0.025) / -0.00043

Using a calculator, we find that t is approximately 5571.85 years.

Answer 2

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( 50)/(2000)=e^(-0.00043t)`

`\Rightarrow( 1)/(400)=e^(-0.00043t)`

Taking ln function both sides of the above equation

`\Rightarrow ln|( 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.