Problem 21 - Calculus Early Transcendentals

Question

Devise an exponential decay function that fits the following data; then answer the accompanying questions. Be sure to identify the reference point \((t=0)\) and units of time. The homicide rate decreases at a rate of \(3 \% / \mathrm{yr}\) in a city that had 800 homicides/yr in \(2010 .\) At this rate, when will the homicide rate reach 600 homicides/yr?

Step-by-Step Solution

Answer

Answer: The homicide rate will reach 600 homicides/year in the year 2020.

Step 1: Formulate the exponential decay function

We can represent the problem using the exponential decay function: \(N(t) = N_0 \times (1 - r)^t\), where \(N(t)\) is the number of homicides after \(t\) years, \(N_0\) is the initial number of homicides, \(r\) is the rate of decrease, and \(t\) is the time in years. For our problem, \(N_0 = 800\), \(r = 0.03\), and \(t = 0\) represents the year 2010. Thus, the function is: \(N(t) = 800 \times (1 - 0.03)^t\).

Step 2: Set up the equation to find the year when the homicide rate reaches 600

We need to find the value of \(t\) when the homicide rate is 600 homicides per year. So, we set \(N(t) = 600\) and solve for \(t\): \(600 = 800 \times (1 - 0.03)^t\).

Step 3: Solve for t

To solve for \(t\), we can divide both sides of the equation by 800 and then take the natural logarithm of both sides: \(\frac{600}{800} = (1 - 0.03)^t\) \(\frac{3}{4} = (0.97)^t\) Taking natural logarithm on both sides: \(\ln(\frac{3}{4}) = t\ln(0.97)\) Now, we can isolate \(t\) by dividing by \(\ln(0.97)\): \(t = \frac{\ln(\frac{3}{4})}{\ln(0.97)} \approx 9.30\)

Step 4: Find the year when the homicide rate reaches 600

Since we consider \(t=0\) as the reference point representing the year 2010, we add the value of \(t\) that we found to 2010: \(2010 + 9.30 \approx 2019.30\) As we cannot have a fraction of a year, we can round up to the next full year: 2020. So, the homicide rate will reach 600 homicides/year in the year 2020.

Key Concepts

Homicide Rate

Understanding the concept of a homicide rate is crucial when dealing with time-dependent changes like exponential decay. The homicide rate typically refers to the number of homicides occurring per unit of time, often expressed as homicides per year in a given population or area. In the context of our problem, the homicide rate started at 800 homicides per year in 2010.
The rate provides a baseline measurement, which can be used to see how effectively prevention and law enforcement measures are. Understanding this rate helps in forecasting future crime patterns and building policies to improve community safety.

Decay Function Formulation

An exponential decay function is often used to model time-dependent processes where the quantity decreases at a proportional rate to its current value.
In this type of function, the rate of decrease is constant, which simplifies modeling and prediction. In the scenario of the decreasing homicide rate, the function is expressed as:

• \(N(t) = N_0 \times (1 - r)^t\)

where:

• \(N(t)\) is the number of homicides after time \(t\),
• \(N_0\) is the initial number of homicides, here 800,
• \(r\) is the decay rate, 0.03 or 3%,
• \(t\) is the time in years.

This function helps us predict changes over time, like when the homicide rate is expected to hit a certain target amount.

Exponential Function Solution

To solve an exponential function, we rearrange it to isolate the variable of interest. Here, we wanted to know when the homicide rate would decrease to 600 homicides per year. This requires solving for \(t\):

• Start with the equation: \(600 = 800 \times (0.97)^t\).
• Simplify to find a ratio: \(\frac{3}{4} = (0.97)^t\), because \(\frac{600}{800} = \frac{3}{4}\).
• Use logarithms to solve for \(t\): \( \ln(\frac{3}{4}) = t \ln(0.97)\).
• Isolate \(t\): \( t = \frac{\ln(\frac{3}{4})}{\ln(0.97)} \approx 9.30\) years.

By solving this function, we determine how long it takes, in this scenario, for the homicide rate to decrease to the desired level. This calculation is essential in predicting future outcomes.

Mathematical Modeling of Time-dependent Processes

Mathematical modeling helps us understand and predict changes within systems over time. In the case of the city’s homicide rate, the model starts from a known point in time and uses a decay function to simulate future declines.
The exponential decay model tells us how long something takes to change at a consistent percentage rate over each time period. This can be critical for planning and decision-making in fields such as public policy, environmental science, or healthcare.

• The model uses past data to project forward, giving insights into when significant milestones or targets will be reached.
• It simplifies complex real-world phenomena, making it easier to understand and work with for strategizing and planning purposes.
• By estimating the year the homicide rate will reach 600, city planners can predict the effects of ongoing or new interventions.

Using such models allows stakeholders to take informed actions based on projected outcomes.