Radioactivity

The formula for radioactive decay relating initial quantity ( ), final quantity ( ), half-life ( ), and time elapsed ( years) is: . Given values: Initial mass . Final mass . Half-life . Time elapsed years. We need to find . Substitute the values into the decay formula: . Divide by 1000 g: . Simplify the fraction : . So, . We know that . Therefore, . Equating the exponents: . Now, solve for : . . This means that after years, the element is left with 125 g. This matches option (a).

The problem revolves around the half-life formula of a radioactive substance. The half-life, denoted as , is the time taken for half of the radioactive atoms to decay. Here, minutes.

Given and are the number of atoms decayed in 20 and 30 minutes respectively. We need to find the ratio .

The formula for the remaining atoms after time is:

Where is the initial number of atoms.

Thus, atoms decayed .

Thus, atoms decayed .

To simplify:

Thus, the ratio is 6:7.