Math Problem Statement

Plutonium is a radioactive atom, which decays at a rate of r=(9.2)⋅10−13. Let p(t) denote the radioactivity at time t ( t is in seconds) and p0 the initial radioactivity. Then p(t) satisfies the decay equation p(t)=p0e−rt.

a) (6 pts) Find the inverse of this function and explain its meaning.

b) (4 pts) How many years does it take for a piece of plutonium of initial radioactivity of p0=100 to decay to 5% of its initial radioactivity?

Solution

Given the problem, we have a radioactive decay equation of the form:

p(t)=p0e−rtp(t) = p_0 e^{-rt} where r=(9.2)×10−13r = (9.2) \times 10^{-13} is the decay rate, p(t)p(t) is the radioactivity at time tt, and ( p_0

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Math Problem Analysis

Mathematical Concepts

Exponential Decay
Radioactive Decay
Inverse Functions
Logarithms

Formulas

p(t) = p_0 e^{-rt}
Inverse: t = -(1/r) ln(p(t)/p_0)

Theorems

Properties of Exponential and Logarithmic Functions

Suitable Grade Level

Undergraduate/Advanced High School

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