A nuclear power plant supplying electrical power to a village uses a radioactive material of half-life T years as the fuel. The amount of fuel at the beginning is such that the total power requirement of the village is 12.5% of the initial power produced by the plant.

Let E be the initial power produced by the plant and E' be the power consumed by the village. We are given that E' = 0.125E (12.5% of E).

The power produced by the plant decreases with time due to radioactive decay. After one half-life (T years), the power output will be E/2. After two half-lives (2T years), the power output will be E/4. After three half-lives (3T years), the power output will be E/8.

We want to find the number of half-lives (n) such that the power produced is at least equal to the power consumed by the village. So we need to solve for n in the equation:

E / 2^n ≥ E'

Substituting E' = 0.125E:

E / 2^n ≥ 0.125E

Dividing both sides by E (assuming E > 0):

1 / 2^n ≥ 0.125

Since 0.125 = 1/8, we have:

1 / 2^n ≥ 1/8

This implies that 2^n ≤ 8. Since 8 = 2^3, we have:

2^n ≤ 2^3

Therefore, n ≤ 3. Thus, the number of half-lives is 3 until the power requirement is met.

Therefore, n = 3