8. (1 point) Household smoke detectors contain of a radioactive element that has a half-life of 432 years. To
   the nearest , how much of this element will remain after 20 years?
   

To find out how much of the radioactive element remains after 20 years, we use the half-life formula. The formula for the remaining amount of a substance over time is given by:

Where:

Here, , years, and years. Plugging in the values:

Calculating the exponent:

Thus,

To the nearest , we find .

The final answer is: c. 190 µg.

Now for some fun extra information!

Radioactive decay is a natural process that occurs at a predictable rate for different isotopes, which is fascinatingly represented through the idea of “half-lives.” These half-lives can span across seconds to billions of years, making the concept both versatile and sometimes mind-boggling! Think about how tiny amounts of certain isotopes can last long enough to help determine the age of archaeological finds!

Did you know that smoke detectors often use Americium-241, a radioactive isotope? The significance of detect potential fires while also employing a material that undergoes radioactive decay for its functionality is a cool intersection of physics and safety! They’re like little guardians of your home, using science to alert you before the fire brigade even breaks a sweat!