Question

An online survival game begins a marathon session with over 65,000 players active on the server. Every hour, the half of the active players whose scores are the lowest get eliminated from the game. If \(g(t)\) is the number of players remaining in the game after \(t\) hours, which of the following best describes the function \(g\) ? A) The function \(g\) increases exponentially. B) The function \(g\) decreases exponentially. C) The function \(g\) increases linearly. D) The function \(g\) decreases linearly.

Short answer

The function \(g(t) = 65,\!000 \cdot (\frac{1}{2})^{t}\) decreases exponentially over time. Therefore, the correct answer is: B) The function \(g\) decreases exponentially.

Step-by-step solution

Identify the initial value

First, let's establish the value of \(g(0)\), which is the initial number of active players (before any eliminations): \[g(0) = 65,\!000\]

Determine the change pattern

Next, we need to observe how the number of active players changes each hour. Since every hour, half of the active players are eliminated, the remaining number of players after each hour is divided by 2. For example: - After 1 hour, the number of active players will be \(g(1) = \frac{1}{2} \cdot g(0)\) - After 2 hours, the number of active players will be \(g(2) = \frac{1}{2} \cdot g(1)\) - And so on... Since the number of active players decreases by half each hour, we can observe that the function \(g\) decreases over time, and the relationship is proportional to the exponent of \(t\), such as: \[g(t) = 65,\!000 \cdot (\frac{1}{2})^{t}\]

Match the function with a given option

Now that we have determined the function \(g(t)\), we can compare it to the options provided: A) The function \(g\) increases exponentially. (False, since \(g(t)\) decreases over time) B) The function \(g\) decreases exponentially. (True, this is the correct description) C) The function \(g\) increases linearly. (False, since \(g(t)\) decreases and it is not linear) D) The function \(g\) decreases linearly. (False, since \(g(t)\) is not linear)

Write the final answer

The correct option that best describes the function \(g(t)\) is: B) The function \(g\) decreases exponentially.

Key concepts

SAT Math Prep

Preparing for the math section of the SAT can be a bit daunting, but understanding various types of functions, especially exponential functions, is a key to success. The SAT often includes scenarios where a quantity increases or decreases by a fixed percentage over time—classic examples of exponential growth and decay. The problem from our exercise is a direct application of an exponential decay scenario, where understanding how to handle exponential functions is crucial.

In preparation for such questions, students should familiarize themselves with the general form of an exponential function, which is typically expressed as: \(f(x) = ab^{x}\), where \(a\) is the initial amount, \(b\) is the base representing growth or decay rate (a value greater than 1 signifies growth, while a value between 0 and 1 signifies decay), and \(x\) is the exponent, which often corresponds to time.

Practicing problems like the one provided helps students recognize patterns of exponential functions, aiding them immensely in test situations where quick identification and analysis are required. Remember, understanding the behavior of the function will lead you to the right answer faster, which is critical in a timed exam like the SAT.

Function Analysis

Function analysis involves examining various characteristics of functions, such as rates of change, intercepts, and long-term behavior. In the context of our exercise, we analyze an exponential function to determine how it behaves over time. As the number of players decreases by a factor of 1/2 each hour, we discern that the function is subject to exponential decay.

An important aspect of function analysis is to understand the distinction between linear and exponential behaviors. A linear function has a constant rate of change and is graphed as a straight line, while an exponential function changes more rapidly, either increasing or decreasing, and its graph curves away from the x-axis.

For exponential functions, the base of the exponent dictates the function’s rate of change. Because the base in our exercise is 1/2, the function decreases by half with each passing hour, resulting in a rapid decrease as time progresses, hence demonstrating exponential decay rather than a linear relationship. This kind of analysis is imperative for correctly interpreting mathematical models and can be widely applied across various fields of study.

Exponential Decay

Exponential decay refers to a process where a quantity decreases by a consistent percentage over equal time intervals. This is exemplified in our exercise by the number of players being halved every hour. In mathematical terms, we described this with the function \(g(t) = 65,000 \times (\frac{1}{2})^{t}\), where \(t\) represents time in hours, and the base \((\frac{1}{2})\) of the exponent signifies the decay factor.

Exponential decay is not just a mathematical concept—it is fundamental in sciences like physics and biology. In the real world, it's seen in phenomena such as radioactive decay, where atoms lose particles and energy over time, or depreciation in finance, where the value of an asset decreases over its useful life.

Understanding exponential decay provides insight into many natural processes and can help in predicting future behavior of decaying quantities; thus, comprehension of this concept is useful not just for academic pursuits but also for practical applications in everyday life.