Question

The amount of carbon- 15 in a given sample decays exponentially with time. If the function \(C(m)=100\left(\frac{1}{2}\right)^{24 m}\) models the amount of carbon-15 remaining in the sample after \(m\) minutes, which of the following must be true? A) The amount of carbon in the sample halves every minute. B) The amount of carbon in the sample halves every 24 minutes. C) The amount of carbon in the sample halves 24 times every minute. D) The amount of carbon in the sample reduces by a factor of 24 every 2 minutes.

Short answer

None of the given statements (A, B, C, D) are true. The correct statement should be: The amount of carbon in the sample halves every 2.5 minutes.

Step-by-step solution

Identify the decay function

The given function representing the decay of carbon-15 is: \[C(m) = 100\left(\frac{1}{2}\right)^{24m}\]

Analyze the exponential term

The exponential term in the function is \(\left(\frac{1}{2}\right)^{24m} \). This term describes how the amount of carbon-15 changes over time. Specifically, the exponent is multiplied by 24, which is an essential factor we need to consider while finding the accurate statement.

Determine the condition when the amount of carbon-15 halves

We need to find when the function C(m) becomes half. Let x be the time when it is half of the initial amount. We can write this equation as: \[50 = 100\left(\frac{1}{2}\right)^{24x}\] Now, to solve for x:

Isolate the exponential term

Divide both sides of the equation by 100: \[\frac{1}{2} = \left(\frac{1}{2}\right)^{24x}\]

Solve for x

Since both sides of the equation have the base 1/2, their exponents must be equal: \[1 = 24x\] Divide by 24: \[x = \frac{1}{24}\] This result shows that the amount of carbon-15 in the sample halves every 1/24 minutes. Now, let's convert these units to better understand the result:

Convert units

As there are 60 minutes in an hour, let's find out how much time it takes for the amount to be reduced by half: \[\frac{1}{24}*60 = 2.5 \text{ minutes}\] Now let's check which of the given statements is true: A) The amount of carbon in the sample halves every minute. B) The amount of carbon in the sample halves every 24 minutes. C) The amount of carbon in the sample halves 24 times every minute. D) The amount of carbon in the sample reduces by a factor of 24 every 2 minutes. Looking at the options, we can conclude that:

Conclusion

None of the given statements (A, B, C, D) are true. The correct statement should be: The amount of carbon in the sample halves every 2.5 minutes.

Key concepts

Exponential Decay

In the context of SAT math practice, understanding exponential decay is critical, especially since it is frequently represented in real-world scenarios, such as radioactive decay or depreciation of assets. The principle of exponential decay is that the quantity of a substance decreases at a rate proportional to its current value. In mathematical terms, this process can be described by a formula like
\[ A(t) = A_0 e^{-kt} \]
where \( A(t) \) represents the amount remaining at time \( t \), \( A_0 \) is the initial amount, \( e \) is the base of the natural logarithm, and \( k \) is the decay constant. The decay constant \( k \) determines the rate at which the quantity decreases over time. In the given SAT math problem, this concept is illustrated through the decay function of carbon-15, where the decay happens every fixed period rather than continuously. The key takeaway is that exponential decay models are not linear, and therefore the amount decreases by a specific percentage over equal time intervals.

Exponential Functions

Exponential functions are fundamental for various fields, from finance to natural sciences. They're characterized by an equation of the form:
\[ f(x) = ab^{cx+d} \]
where \( a \), \( b \), \( c \), and \( d \) are constants, and \( b \) is the base of the exponential. In our SAT math example, the function \( C(m) = 100\left(\frac{1}{2}\right)^{24m} \) dictates how the amount of a substance changes over time — in this case, the percentage of carbon-15 left after \( m \) minutes. The base here is \(\frac{1}{2}\) indicating that the remaining amount is halving over certain intervals. Exponential functions often involve growth or decay, and on the SAT, students should be adept at identifying and manipulating these functions to solve problems.

Math Problem Solving

Math problem solving is a skill that encompasses understanding the problem, devising a plan, carrying out that plan, and finally checking the results. In solving an SAT math problem involving exponential decay, students must first identify the correct model—such as an exponential decay function—and then analyze and manipulate the function to find the required answer. For example, in our problem with carbon-15 decay, the calculation process requires isolating the exponential term and understanding the significance of the variables and constants within the equation. Throughout the steps, students must maintain a logical approach, ensuring that each step is purposeful and brings them closer to the correct solution. It's crucial to interpret results and units correctly, as demonstrated when converting time units from minutes to a larger or smaller unit to match the answer choices given in the problem.

SAT Prep

Effective SAT preparation involves developing a clear understanding of key concepts and question formats across subjects, including math. When studying exponential functions and decay for the SAT math section, students should:

• Memorize the general form of exponential functions.
• Practice with different forms of exponential equations—some may represent growth and others decay.
• Learn to quickly identify the base and exponent in an equation and understand their role.
• Work on converting units, as many problems require answers in different units than those given in the question.
• Recognize common traps in answer choices and how to avoid them.

Every practice problem tackled enhances understanding and speed, both of which are essential to perform well in the actual SAT exam. Regular practice with a variety of problems, such as the given carbon-15 decay example, can help students become proficient in recognizing patterns, applying formulas correctly, and confidently navigating the test.