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Chapter 3: Problem 29

Use the exponential decay model, \(A=A_{0} e^{k t},\) to solve Exercises \(28-31 .\) Round answers to one decimal place. The half-life of lead is 22 years. How long will it take for a sample of this substance to decay to \(80 \%\) of its original amount?

Short Answer

Expert verified
It would take approximately 15 years for the lead to decay to 80% of its original amount.

Step by step solution

01

Calculation of decay constant \(k\)

The decay constant, \(k\), can be obtained from the concept of half-life. It is the value of \(k\) in the equation \(A_{0}/2=A_{0} e^{22k}\). By solving this equation, we get the value of \(k\) as \(k=\ln(0.5)/-22\), where \(ln\) denotes the natural logarithm.
02

Finding the decay time to reach 80% of original value

After finding the decay constant, replace \(A\) with \(0.8A_{0}\) in the decay equation to get \(0.8A_{0}=A_{0} e^{kt}\). To solve for \(t\), divide both sides of the equation by \(A_{0}\), which gives \(0.8=e^{kt}\). Take the natural logarithm of both sides to get the equation \(ln(0.8)=kt\). Dividing both sides by \(k\) gives \(t=ln(0.8)/k\). Remember that \(k=\ln(0.5)/-22\) from Step 1.
03

Solving for the decay time

Substitute the value of \(k\) into the equation from Step 2 to get \(t=-ln(0.8)*(-22)/\ln(0.5)\). Completing this calculation gives the decay time. Your answer should be rounded to the nearest tenth.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Half-Life Calculation
The half-life of a substance is the time required for half of the initial amount of the substance to decay. Understanding half-life is crucial in fields like archaeology, medicine, and environmental science, where it's used to date artifacts, trace the breakdown of pharmaceuticals, and monitor the decay of pollutants, respectively. Let's break this down with respect to our lead example from the exercise.

In this scenario, the half-life of lead is given as 22 years. This means that every 22 years, the amount of lead will be reduced to half of its initial value. When students need to calculate how long it will take for a certain amount of substance to decay to a particular percentage of its original amount, they must modify the traditional half-life calculation to accommodate the desired final proportion.
Decay Constant
The decay constant, denoted by the symbol \( k \), signifies the rate at which a substance undergoes exponential decay. It's inherently linked to the half-life of the substance鈥攖he smaller the decay constant, the slower the rate of decay, and hence, the longer the half-life. For the exercise involving lead, we utilize the half-life to determine the decay constant.

The equation \( k = \frac{\ln(0.5)}{-22} \) reveals that the decay constant is calculated using the natural logarithm of one-half, which corresponds to the half-life decay, divided by the half-life period in years. This calculated value of \( k \) is crucial to figure out the decay over time periods different from the half-life, such as the 80% decay milestone in our example.
Natural Logarithm
The natural logarithm is the logarithm to the base \( e \) (where \( e \) is an irrational constant approximately equal to 2.71828). It is great for solving equations where the unknown variable is an exponent, which is common in growth and decay problems. For instance, in our lead decay problem, we used the natural logarithm to isolate \( t \) and solve for the time it takes to reach 80% of the original amount.

The step that involves \( \ln(0.8) = kt \) is facilitated by the properties of logarithms, which allow us to 'bring down' the exponent and solve for \( t \) directly. Understanding the nature of the natural logarithm, particularly its role in undoing the exponential function, is pivotal in calculating decay time in exponential decay models.

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Most popular questions from this chapter

The \(p H\) scale is used to measure the acidity or alkalinity of a solution. The scale ranges from 0 to \(14 .\) A neutral solution, such as pure water, has a pH of 7. An acid solution has a pH less than 7 and an alkaline solution has a pH greater than 7. The lower the \(p H\) below 7 , the more acidic is the solution. Each whole-number decrease in \(p H\) represents a tenfold increase in acidity. (GRAPH CAN'T COPY). The \(p H\) of a solution is given by $$\mathrm{pH}=-\log x$$ where \(x\) represents the concentration of the hydrogen ions in the solution, in moles per liter. Use the formula to solve. Express answers as powers of \(10 .\) a. Normal, unpolluted rain has a pH of about 5.6. What is the hydrogen ion concentration? b. An environmental concern involves the destructive effects of acid rain. The most acidic rainfall ever had a \(\mathrm{pH}\) of \(2.4 .\) What was the hydrogen ion concentration? c. How many times greater is the hydrogen ion concentration of the acidic rainfall in part (b) than the normal rainfall in part (a)?

Approximate each number using a calculator. Round your answer to three decimal places. $$2^{3.4}$$

Use Newton's Law of Cooling, \(T=C+\left(T_{0}-C\right) e^{k t},\) to solve Exercises \(47-50\). A frozen steak initially has a temperature of \(28^{\circ} \mathrm{F}\). It is left to thaw in a room that has a temperature of \(75^{\circ} \mathrm{F}\). After 10 minutes, the temperature of the steak has risen to \(38^{\circ} \mathrm{F}\). After how many minutes will the temperature of the steak be \(50^{\circ} \mathrm{F} ?\)

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$\log _{3}(4 x-7)=2$$

Use Newton's Law of Cooling, \(T=C+\left(T_{0}-C\right) e^{k t},\) to solve Exercises \(47-50\). A pizza removed from the oven has a temperature of \(450^{\circ} \mathrm{F}\) It is left sitting in a room that has a temperature of \(70^{\circ} \mathrm{F}\). After 5 minutes, the temperature of the pizza is \(300^{\circ} \mathrm{F}\) a. Use Newton's Law of Cooling to find a model for the temperature of the pizza, \(T\), after \(t\) minutes. b. What is the temperature of the pizza after 20 minutes? c. When will the temperature of the pizza be \(140^{\circ} \mathrm{F} ?\)
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