Question

Designing exponential decay functions 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. Crime rate The homicide rate decreases at a rate of \(3 \% / \mathrm{yr}\) in a city that had 800 homicides/yr in 2018 . At this rate, when will the homicide rate reach 600 homicides/yr?

Short answer

Answer: We can expect the homicide rate to reach 600 homicides/yr in the year 2027.

Step-by-step solution

Understand the Exponential Decay Function

An exponential decay function is a model used to describe a quantity decreasing by a constant percentage, and it has the form: $$f(t) = f_0 \cdot (1-r)^t$$ Where \(f(t)\) is the quantity at a particular time \(t\), \(f_0\) is the initial value, \(r\) is the rate of decay, and \(t\) is the time.

Fill in the given information

We know that the initial homicide rate was 800 homicides/yr in 2018, which will be our reference point, \((t=0)\). And the decay rate is \(3\% = 0.03\) per year. Now, we can fill in the information into our function: $$f(t) = 800 \cdot (1 - 0.03)^t$$

Solve for the time when the homicide rate is 600 homicides/yr

We need to find the time \(t\) when the homicide rate \(f(t)\) reaches 600 homicides/yr. So, we set up the equation: $$600 = 800 \cdot (1 - 0.03)^t$$

Solve for \(t\)

First, divide both sides by 800: $$\frac{600}{800} = (1 - 0.03)^t$$ Simplify: $$0.75 = (0.97)^t$$ Now, to solve for \(t\), we can take the logarithm of both sides (using any base): $$\log{(0.75)} = \log{(0.97)^t}$$ Using the logarithm property \(\log{(a^b)} = b \cdot \log{(a)}\): $$\log{(0.75)} = t \cdot \log{(0.97)}$$ Now, divide both sides by \(\log{(0.97)}\): $$t = \frac{\log{(0.75)}}{\log{(0.97)}}$$ Calculate the value of \(t\): $$t \approx 8.92$$

Interpret the result

The value of \(t \approx 8.92\) means that, at the given rate of decay, the homicide rate in the city will reach 600 homicides/yr after approximately 8.92 years. Since the reference year was 2018, we can expect that the homicide rate will be 600 homicides/yr around the year 2027.

Key concepts

Calculus

Calculus, the mathematical study of continuous change, plays a vital role in understanding the behavior of functions and modeling real-world scenarios. It consists of two primary branches: differential calculus, concerning rates of change and slopes of curves, and integral calculus, which focuses on accumulation of quantities and areas under curves. Exponential decay functions often require the application of calculus for solving various problems, including determining rates of change and predicting future values.

For example, calculus helps us compute the precise time it will take for a certain quantity to diminish to a particular level under exponential decay, as seen in the homicide rate scenario. Using differentiation, one can find the instantaneous rate of decay at any point, and with integration, we can ascertain the accumulated decrease over a period.

Exponential Decay Model

The exponential decay model is integral to representing quantities that decrease at a rate proportional to their current value. This model is widely used in fields like physics, chemistry, finance, and even criminology for analyzing decline trends. The form of an exponential decay function is characterized by the equation \(f(t) = f_0 \cdot (1-r)^t\), where:\

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• \(f(t)\) represents the quantity at time \(t\)
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• \(f_0\) is the initial quantity
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• \(r\) is the decay rate (expressed as a fraction)
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• \(t\) represents time
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In the exercise provided, we apply this model to the real-world situation of a decreasing crime rate. This practical application allows us to project into the future and predict when the homicide rate reaches a certain threshold. Understanding the exponential decay model is pivotal in interpreting these situations and making informed decisions or predictions based on the data.

Logarithmic Equations

Logarithmic equations are essential tools for solving problems involving exponential functions. A logarithm is the inverse operation to exponentiation, answering the question 'to what exponent must a base be raised, to produce a certain number?'. The equation \(y = log_b(x)\) means the same as \(b^y = x\), where \(b\) is the base of the logarithm.

When dealing with exponential decay, logarithms allow us to solve for time or any other exponent in our function. As demonstrated in our homicide rate example, once we established the equation \(600 = 800 \cdot (1 - 0.03)^t\), we used logarithmic equations to solve for \(t\). This particular approach is efficient for determining the period it will take for a decaying quantity to reach a specific value and is an excellent example of the practical use of logarithmic equations in analyzing real-world exponential decay problems.