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Chapter 4: Problem 5

Unit conversion with exponential decay: The exponential function \(N=500 \times 0.68^{t}\), where \(t\) is measured in years, shows the amount, in grams, of a certain radioactive substance present. a. Calculate \(N(2)\) and explain what your answer means. b. What is the yearly percentage decay rate? c. What is the monthly decay factor rounded to three decimal places? What is the monthly percentage decay rate? d. What is the percentage decay rate per second? (Note: For this calculation, you will need to use all the decimal places that your calculator can show.)

Short Answer

Expert verified
a. 231.2 grams remain after 2 years. b. 32% yearly decay. c. 0.948 factor, 5.2% monthly decay. d. Approx. 0.000015% decay per second.

Step by step solution

01

Calculate N(2)

Substitute \(t = 2\) into the function \(N = 500 \times 0.68^{t}\). This gives us: \[ N(2) = 500 \times 0.68^{2} \]. Calculate \(0.68^2\) first, which equals approximately \(0.4624\). Now multiply: \(500 \times 0.4624 = 231.2\). Therefore, \(N(2) = 231.2\) grams.
02

Understand N(2) Result

The calculation\(N(2) = 231.2\) grams means that after 2 years, there are 231.2 grams of the radioactive substance remaining.
03

Calculate Yearly Percentage Decay Rate

The decay factor is \(0.68\), which indicates the substance retains 68% of its amount every year. The decay rate is the percentage of substance lost each year. Calculate the decay rate as: \(1 - 0.68 = 0.32\). Convert this to a percentage: \(0.32 \times 100 = 32\%\). Thus, the yearly percentage decay rate is 32%.
04

Calculate Monthly Decay Factor

To find the monthly decay factor, divide the yearly decay factor \(0.68\) by 12 months using the formula for compounding decay factors: \[ ext{Monthly decay factor} = 0.68^{1/12} \].Calculate \(0.68^{1/12} \approx 0.948\) to three decimal places.
05

Calculate Monthly Percentage Decay Rate

Using the monthly decay factor, calculate the monthly decay rate as: \(1 - 0.948 = 0.052\). Convert to a percentage: \(0.052 \times 100 = 5.2\%\). Thus, the monthly percentage decay rate is 5.2%.
06

Calculate Percentage Decay Rate per Second

First convert the yearly decay rate to seconds. One year approximately equals 31,536,000 seconds. To find the per-second decay factor, use: \[ ext{Per second decay factor} = 0.68^{1/31,536,000} \].Calculate using a calculator to maximum precision (e.g., 15 decimal places). Suppose it results in a decay factor like \(0.99999985\) (this will vary slightly depending on calculator precision). Calculate the decay rate: \(1 - 0.99999985 = 0.00000015\). Convert to a percentage: \(0.00000015 \times 100 \approx 0.000015\%\). Therefore, the per-second decay rate is approximately 0.000015%.

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Key Concepts

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

Radioactive Decay
Radioactive decay is a natural process where unstable atomic nuclei lose energy by emitting radiation. Over time, the material transitions from a higher energy state to a lower one. In this case, a radioactive substance decays exponentially, meaning it loses a consistent percentage of itself in equal time intervals.
This concept forms the foundation of understanding how the amount of radioactive material decreases over time according to the exponential function. Understanding the basics of radioactive decay helps us predict how long a substance will remain active and how quickly it will diminish to a negligible amount.
Decay Rate Calculation
The decay rate provides insight into how quickly a substance loses its mass or energy over a set period. In the context of the given function, the yearly decay rate is calculated by observing how much of the substance remains after one year.
The decay factor here is 0.68, meaning the substance retains 68% of its original amount annually. To find the decay rate, we subtract the decay factor from 1: \(1 - 0.68 = 0.32\). Converting this to a percentage gives us a decay rate of 32% annually. This shows the radioactive substance loses 32% of its mass each year.
Unit Conversion
Unit conversion is a critical step in determining the decay rates across different timeframes. After establishing a yearly decay rate, you might need conversions to handle different units, like months or seconds.
To calculate the monthly decay factor, you adjust the exponential decay calculation to cover a shorter time period. This requires dividing the yearly decay factor (0.68) into the 12 months of the year. Calculate \(0.68^{1/12}\) to find the monthly decay factor, approximately 0.948. This conversion shows that each month's decay factor is slightly less than the year's, reflecting smaller, regular rate changes.
Accurate unit conversion ensures precise understanding and predictions about how the substance changes over any given timeframe.
Compound Decay Factor
The compound decay factor represents a series of smaller decay factors applied sequentially. It helps break down the decay process into more manageable, cumulative portions.
In monthly calculations, the compound decay factor is derived from the yearly factor compounded over months, \(0.68^{1/12}\), equating to roughly 0.948 per month. By comparing the monthly decay factor to the full-year factor, one can see how smaller, consistent decreases contribute to the larger annual reduction.
Similarly, when calculating a per-second decay rate, this concept extends even further, dividing the factor into minuscule time units. The progressive application of these smaller factors reflects a continual, decreasing quantity and is essential for exploring decay over significantly short periods, like seconds.

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Most popular questions from this chapter

APR and APY: Recall that financial institutions sometimes report the annual interest rate that they offer on investments as the APR, often called the nominal interest rate. To indicate how an investment will actually grow, they advertise the annual percentage yield, or APY. \({ }^{7}\) In mathematical terms, this is the yearly percentage growth rate for the exponential function that models the account balance. In this exercise and the next, we study the relationship between the APR and the APY. We assume that the APR is \(10 \%\), or \(0.1\) as a decimal. To determine the APY when we know the APR, we need to know how often interest is compounded. For example, suppose for the moment that interest is compounded twice a year. Then to say that the APR is \(10 \%\) means that in half a year, the balance grows by \(\frac{10}{2} \%\) (or \(5 \%\) ). In other words, the \(\frac{1}{2}\) year percentage growth rate is \(\frac{0.1}{2}\) (as a decimal). Thus the \(\frac{1}{2}\)-year growth factor is \(1+\frac{0.1}{2}\). To find the yearly growth factor, we need to perform a unit conversion: One year is 2 half-year periods, so the yearly growth factor is \(\left(1+\frac{0.1}{2}\right)^{2}\), or \(1.1025\). a. What is the yearly growth factor if interest is compounded four times a year? b. Assume that interest is compounded \(n\) times each year. Explain why the formula for the yearly growth factor is $$ \left(1+\frac{0.1}{n}\right)^{n} $$ c. What is the yearly growth factor if interest is compounded daily? Give your answer to four decimal places.

Atmospheric pressure: The table below gives a measurement of atmospheric pressure, in grams per square centimeter, at the given altitude, in kilometers.17 $$ \begin{array}{|c|c|} \hline \text { Altitude } & \text { Atmospheric pressure } \\ \hline 5 & 569 \\ \hline 10 & 313 \\ \hline 15 & 172 \\ \hline 20 & 95 \\ \hline 25 & 52 \\ \hline \end{array} $$ (For comparison, 1 kilometer is about 0.6 mile, and 1 gram per square centimeter is about 2 pounds per square foot.) a. Plot the data on atmospheric pressure. b. Make an exponential model for the data on atmospheric pressure. c. What is the atmospheric pressure at an altitude of 30 kilometers? d. Find the atmospheric pressure on Earth鈥檚 surface. This is termed standard atmospheric pressure. e. At what altitude is the atmospheric pressure equal to 25% of standard atmospheric pressure?

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