"In a bad late-night science fiction film, a villain is using a large collection of radioactive atoms as energy for a weapon to threaten the good guys. The atoms have a half-life of 1 hour. The villain has 4 kilograms of the radioactive material now, and he needs a minimum of 1 kg. for his weapon to work. After how much time will the weapon no longer be a threat?"

t=1.4hours

Explanation:

The half life is 1hour

At t=0 he has a mass of 4kg

So he want it to be 1kg, so that his weapon can work.

Applying the exponential function of decay

M=Cexp(-kt)

Where,

M is the mass at any time

C is a constant of integration

k is the rate of decay

Given that it has an half life of 1 hours.

Then k is 1

At t =0 the mass is 4kg

Therefore

4=Cexp(0)

C=4

M=4exp(-kt)

Since rate of decay is 1, then k=1

M=4exp(-t)

We need to find t at M=1kg

1=4exp(-t)

1=4exp(-t)

1/4=exp(-t)

0.25=exp(-t)

Take In of both sides

In(0.25)=-t

-1.3863=-t

Then, t=1.386hour

Then it will take about 1.4 hours to get to 1kg.

it about 600 mph from what i calculated

r=rho l/a where a is the area, l is the length and rho is resistivity so the greater the area the lesser the resistance thats why the earth pin is thicker bcos current follow the shortest path, so less resistive .