Problem 16
Question
An artifact originally had 16 grams of carbon- 14 present. The decay model \(A=16 e^{-0.0001211}\) describes the amount of carbon-I4 present after t years. Use this model to solve Exercises \(15-16 .\) How many grams of carbon-14 will be present in \(11,430\) years?
Step-by-Step Solution
Verified Answer
Approximately 4.01 grams of carbon-14 will be present after 11,430 years.
1Step 1: Identify values
The given decay model is \(A=16 e^{-0.0001211t}\), where 'A' represents the amount of carbon-14 present after t years. 't' is the value for the number of years that have elapsed, which in this exercise is given as 11,430.
2Step 2: Substitute the value of t
Replace the 't' in the decay model with 11,430 to get \(A = 16 e^{-0.0001211 \cdot 11,430}\)
3Step 3: Solve the equation
Solve the equation to find the remaining amount of carbon-14. \(A = 16e^{-1.382993}\). This gives us approximately \(A = 16 \cdot 0.250818\)
4Step 4: Compute the final value
Multiply 16 and 0.250818, you get that \(A \approx 4.01309\) grams. Therefore, approximately 4.01 grams of carbon-14 will be present after 11,430 years.
Key Concepts
Carbon-14 DecayDecay ModelHalf-Life Calculation
Carbon-14 Decay
Carbon-14 decay is a process where the radioactive isotope carbon-14 transforms into nitrogen-14 over time. This transformation is due to beta decay, a type of radioactive decay where a neutron in the carbon-14 nucleus is converted into a proton, releasing an electron (beta particle) and an antineutrino. For scientists, carbon-14 decay is crucial as it serves as a clock to date organic materials that are up to about 50,000 years old. This method of dating, known as radiocarbon dating, relies on measuring the remaining amount of carbon-14 in a sample and using it to estimate the age of the artifact. As carbon-14 decays, the amount present decreases exponentially, following an exponential decay function which provides a predictable way to calculate the age of ancient organic artifacts.
Decay Model
A decay model like the one in this exercise is a mathematical representation that describes how a quantity decreases over time. For carbon-14, the decay model uses an exponential function, which involves an initial quantity and an exponential decay factor.
In the given model, the formula is:
In the given model, the formula is:
- \( A = 16 e^{-0.0001211t} \)
- 'A' represents the amount of carbon-14 remaining after 't' years.
- '16' is the initial amount of carbon-14 in grams.
- 'e' is the base of the natural logarithm, approximately equal to 2.71828.
- '-0.0001211' is the decay constant, specific to carbon-14, which determines the rate at which the substance decreases.
Half-Life Calculation
Half-life is the time required for half of the original quantity of a substance to decay. For carbon-14, this is approximately 5,730 years. The concept of half-life is essential for understanding how quickly a substance undergoes exponential decay.
To calculate the half-life or use it in decay problems, it's important to understand that with each passing half-life, the remaining quantity of the substance is halved.
In exercises using decay models, knowing the half-life allows us to verify the decay constant or vice versa. The half-life is calculated from the decay constant using the formula:
To calculate the half-life or use it in decay problems, it's important to understand that with each passing half-life, the remaining quantity of the substance is halved.
In exercises using decay models, knowing the half-life allows us to verify the decay constant or vice versa. The half-life is calculated from the decay constant using the formula:
- \( ext{Half-life} = \frac{ ext{ln}(2)}{k} \)
- where \( k \) is the decay constant, and ln(2) is the natural logarithm of 2, approximately 0.693.
Other exercises in this chapter
Problem 15
Graph each function by making a table of coordinates. If applicable, use a graphing utility to confirm your hand-drawn graph. $$h(x)=\left(\frac{1}{2}\right)^{x
View solution Problem 15
Use properties of logarithms to expand logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator.
View solution Problem 16
Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$7^{\frac{x-2}{6}}=\sqrt{7}$$
View solution Problem 16
Write each equation in its equivalent logarithmic form. $$15^{2}=x$$
View solution