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Chapter 5: Problem 10

The per-capita consumption of bottled water was 8 gallons in 1990 and has been increasing yearly by a factor of \(1.088 .\) What was the per capita consumption of bottled water 12 years later?

Short Answer

Expert verified
21.752 gallons

Step by step solution

01

Identify the Variables

We need to determine the per capita consumption of bottled water after 12 years. Here, the initial consumption (in 1990) is 8 gallons, the yearly increase factor is 1.088, and the number of years is 12.
02

Formulate the Exponential Growth Equation

The consumption of bottled water increases exponentially. The general formula for exponential growth is \[ P(t) = P_0 \times (growth \text{ factor})^t \]where \( P(t) \) is the consumption after time \( t \), \( P_0 \) is the initial consumption, and \( t \) is the number of years.
03

Substitute the Values into the Equation

Substitute the given values into the exponential growth equation:\[ P(12) = 8 \times 1.088^{12} \]
04

Calculate the Power Term

First, calculate the power term:\[ 1.088^{12} \approx 2.719\]
05

Calculate the Final Consumption

Now multiply the initial consumption by the calculated power term:\[ P(12) eq 8 \times 2.719 \approx 21.752 \] Hence, the per capita consumption of bottled water 12 years later is approximately 21.752 gallons.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

per capita consumption
Per capita consumption refers to the average amount of a particular product or resource consumed per person within a specified period. This measurement helps in understanding how consumption patterns change over time. For example, if we're looking at bottled water consumption, per capita consumption would be the total amount of bottled water consumed divided by the number of people. This metric is particularly useful for identifying trends in consumption, making comparisons between different regions, and for policy-making purposes.
bottled water consumption
Bottled water consumption is the amount of bottled water consumed by individuals over a certain period. This is a popular consumer behavior trend, often measured to understand hydration habits and market needs. For instance, the exercise mentions that in 1990, people consumed 8 gallons of bottled water per person. Monitoring this consumption can help companies plan their production and marketing strategies. It also provides insights for environmental studies, since bottled water production and disposal have significant ecological impacts.
exponential growth equation
The exponential growth equation is used to model scenarios where a quantity increases over time at a rate proportional to its current value. The general formula is:\[ P(t) = P_0 \times (growth \text{ factor})^t \]Here,
  • \( P(t) \) is the amount after time \( t \)
  • \( P_0 \) is the initial amount
  • Growth factor is the rate at which the quantity is increasing annually
  • \( t \) is the time in years
In the given exercise, we're using this equation to determine the per capita consumption of bottled water after 12 years. An exponential growth pattern shows that the consumption doesn't increase linearly but accelerates over time.
initial value
The initial value, or \( P_0 \), is the starting point in calculations involving exponential growth. In our case study, the initial value is the per capita consumption of bottled water in 1990, which is 8 gallons. This starting point is crucial, as it sets the baseline against which all future growth is measured. When solving problems involving exponential growth, identifying the initial value accurately is the first step. It ensures that the calculations reflect the true nature of the growth pattern.
growth factor
The growth factor is a multiplier that reflects how much a quantity increases over a specified period, usually one year. In our example, the growth factor is 1.088, meaning the per capita consumption of bottled water increases by 8.8% annually. The calculation of future values involves raising this factor to the power of the number of years (\( t \)) to see how the consumption evolves. Understanding the growth factor is essential because it encapsulates the rate at which the quantity is changing. Small changes in the growth factor can lead to significant differences over time, highlighting the importance of accurate estimates.

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Most popular questions from this chapter

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