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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 5: Q. 21 (page 351)

A 50-mg sample of a radioactive substance decays to 34 mg after 30 days. How long will it take for there to be 2 mg remaining?

Short Answer

Expert verified

The time taken is approximately 250.5 days.

Step by step solution

01

Given Information

It is given that:

Initial amount is 50mg.

Amount after 30 days is 34mg.

Amount to be determined when it will be 2mg.

02

Initial equation and assumption

It is assumed that the radioactive substance decay is exponential, therefore the equation can be expressed as:

$A_{t} = A_{0} e^{b t}$

where A_t is the amount after time t, A₀ is initial amount and b is time factor.

03

Determining time factor

As per the question,

A₀ = 50mg, A₃₀ = 34mg

Using the formula,

$\begin{array}{rcl} A_{30} & = & A_{0} e^{b (30)} \\ 34 & = & 50 e^{30 b} \\ \frac{34}{50} & = & e^{30 b} \end{array}$

Taking log both sides

$\begin{array}{rcl} 30 b & = & \ln (\frac{17}{25}) \\ b & = & \frac{1}{30} \ln (\frac{17}{25}) \\ b & = & - 0.01285 \end{array}$

04

Calculating the time required

Let the time to be determined is t days

Given A_t = 2mg

Putting it in the equation we get,

$\begin{array}{rcl} A_{t} & = & A_{0} e^{b t} \\ 2 & = & 50 e^{- 0.01285 脳 t} \\ \frac{2}{50} & = & e^{- 0.01285 t} \end{array}$

Taking log on both sides

$- 0.01285 t = \ln (\frac{1}{25}) t = - \frac{1}{0.01285} 脳 - 3.21887 t 鈮� 250.5 days$

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