The half-life of nickel-63 is 100 years. If a sample of nickel-63 decays until 6.25% of the original sample remains, how much time has passed? *

6.25 years
100 years
400 years
1600 years

400 years

Explanation:

The equation that describes the decay of a radioactive sample is:

(1)

where

m(t) is the amount of sample left at time t

is the initial amount of the sample

is the half-life, which is the time taken for the sample to halve

In this problem we have:

is the half-life of Nickel-63

After a time t, the amount of sample left is 6.25% of the original one, which means that

So we can rewrite the equation (1) and solving for t to find the time:

russian is not my first language, so i'm afraid i can't write out an answer in russian, but hopefully you can find a way to do so. here's my best translation of the problem: you're given a plot of a particle's position in one-dimensional space over time, and you're asked to find either the total distance traveled by the particle after 6 seconds had elapsed, or the total displacement of the particle after 6 seconds.

if you want the total distance traveled, you would first need to determine the particle's velocity as a function of time, then integrate that over the 6-second interval. we know the position function is quadratic with roots at 0 and 6, so we have

where is unknown. because the position is parabolic, there is symmetry about the midpoint at , and we know that

so that the position function is

which means the velocity is

then the total distance traveled is

so the total distance traveled is 20 m.

on the other hand, if you want to find displacement: the particle starts at and ends at , so its displacement is 0.