Question

The half-life of a radioactive element is 1,200 years. How old is the object if only 12.5% of the radioactive atoms remain?

Answer 1

3,600 years old

Step-by-step explanation:

To determine the age of the object based on the remaining radioactive atoms, we can use the concept of half-life.

The half-life is the time it takes for half of the radioactive atoms in a sample to decay. In this case, the half-life of the radioactive element is 1,200 years. This means that after 1,200 years, half of the radioactive atoms will have decayed, and the remaining half will still be radioactive.

Half-life formula

`\boxed{N(t)=N_0\left((1)/(2)\right)^{(t)/(t_(1/2))}}`

where:

• N(t) is the quantity remaining.
• N₀ is the initial quantity.
• t is the time elapsed.
• 
  `t_(1/2)` is the half-life of the substance.

The initial quantity is 100% and the quantity remaining is 12.5%. The half-life is 1,200 years. Therefore, the values to substitute into the half-life formula are:

• 
  `N_0=100\%=1`
• 
  `N(t)=12.5\%=0.125`
• 
  `t_(1/2)=1200`

Substitute the values into the half-life formula:

`0.125=1\left((1)/(2)\right)^{(t)/(1200)}`

Solve for t:

`\begin{aligned}0.125&=\left((1)/(2)\right)^{(t)/(1200)}\\\\(1)/(8)&=\left((1)/(2)\right)^{(t)/(1200)}\\\\\ln \left((1)/(8)\right)&=\ln \left((1)/(2)\right)^{(t)/(1200)}\\\\\ln \left((1)/(8)\right)&=(t)/(1200)\ln \left((1)/(2)\right)\\\\1200\ln \left((1)/(8)\right)&=t\ln \left((1)/(2)\right)\\\\1200(\ln1-\ln8)&=t(\ln1-\ln 2)\\\\1200(-\ln8)&=t(-\ln 2)\\\\1200(-\ln2^3)&=t(-\ln 2)\\\\1200(-3\ln2)&=t(-\ln 2)\\\\-3600\ln2&=-t\ln 2\\\\\end{aligned}`

`\begin{aligned}t\ln 2&=3600\ln2\\\\t&=3600\end{aligned}`

Therefore, if only 12.5% of the radioactive atoms remain, the object is approximately 3,600 years old.