Question

I need help

Element X is a radioactive isotope such that every 27 years, its mass decreases by half. Given that the initial mass of a sample of Element X is 7900 grams, how much of the element would remain after 12 years, to the nearest whole number?

Answer 1

About 5, 805 grams remaining.

Explanation:

We are given that every 27 years, the mass of Element X decreases by half.

We can write an exponential function to model the situation. The standard exponential function is given by:

→ f(t)=a(r)^t

Where a is the initial amount, r is the rate, and t is the time, in this case years.

Since it is halved, our rate r is 1/2.

Since it is halved for every 27 years, for t, it will be t/27.

Therefore, our function is:

→ f(t)=a(1/2)^t/27

Our initial sample is 7,900 grams. Hence, a = 7900:

→ f(t)=7900(1/2)^t/27

We want to find the remaining amount after 12 years. So, t = 12. Use a calculator:

→ f(12)=7900(1/2)^12/27 ≈5805 grams

After 12 years, there will be about 5805 grams remaining.

Answer 2

About 5, 805 grams remaining.

Explanation:

We are given that every 27 years, the mass of Element X decreases by half.

We can write an exponential function to model the situation. The standard exponential function is given by:

`\displaystyle f(t)=a(r)^t`

Where a is the initial amount, r is the rate, and t is the time, in this case years.

Since it is halved, our rate r is 1/2.

Since it is halved for every 27 years, for t, it will be t/27.

Therefore, our function is:

`\displaystyle f(t)=a\Big((1)/(2)\Big)^(t/27)`

Our initial sample is 7,900 grams. Hence, a = 7900:

`\displaystyle f(t)=7900\Big((1)/(2)\Big)^(t/27)`

We want to find the remaining amount after 12 years. So, t = 12. Use a calculator:

`\displaystyle f(12)=7900\Big((1)/(2)\Big)^(12/27)\approx5805\text{ grams}`

After 12 years, there will be about 5805 grams remaining.