   Chapter 4.6, Problem 18E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
26 views

# Radioactive Decay Find the half-life of a radioactive material for which 99.57% of the initial amount remains after 1 year.

To determine

To calculate: The half-life of a radioactive material for which 99.57% of initial amount remains after 1 year.

Explanation

Given Information:

The provided information is that 99.57% of initial amount remains after 1 year.

Formula used:

Exponential growth and decay:

If the rate of change of a positive quantity y with respect to time is proportional to the amount of quantity present at any time t, that is dydt=ky, then y is given by the equation, y=Cekt, where C is the value of the quantity at time t=0 and k is the constant of proportionality.

If k>0 then there is exponential growth and when k<0 then there is exponential decay.

Percentage of the remaining amount =Amount leftInitial amount×100

Calculation:

Consider the provided information that 99.57% of initial amount remains after 1 year.

Let C be the initial amount of the radioactive material and h be the half-life of the radioactive material.

As 99.57% of initial amount remains after 1 year, so at t=1 the remaining amount is,

y=99.57% of C=99.57100×C=0.9957×C=0.9957C

Substitute t=1 and y=0.9957C in the equation y=Cekt.

0.9957C=Cek10.9957C=Cek

Also, the half-life of the radioactive material is h, that is, at t=h the initial amount reduces to half so,

y=C2

Substitute t=h and y=C2 in the equation y=Cekt

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

#### let f(x) = x 1, g(x) = x+1, and h(x) = 2x3 1. Find the rule for each function. 14. hg

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

#### In Problems 7-34, perform the indicated operations and simplify. 31.

Mathematical Applications for the Management, Life, and Social Sciences

#### Sometimes, Always, or Never: limxax=0.

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 