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Chapter 3: Problem 92

The total amount spent on analog TVs in the United States changed from 5836 million dollars in 2003 to 1424 million dollars in \(2006 .\) Find and interpret the average rate of change in sales, in millions of dollars per year. Round your answer to the nearest hundredth. (Source: Consumer Electronics Association.)

Short Answer

Expert verified
The average rate of change in sales is approximately -1470.67 million dollars per year.

Step by step solution

01

Understand the Problem

Determine the initial and final values, as well as the time period over which the changes occurred. The initial value (2003) is 5836 million dollars and the final value (2006) is 1424 million dollars. The time period is from 2003 to 2006, which is 3 years.
02

Calculate the Change in Sales

Subtract the final value from the initial value to find the change in sales: \[ \text{Change in sales} = 1424 - 5836 = -4412 \text{ million dollars} \]
03

Calculate the Average Rate of Change

Divide the change in sales by the number of years to find the average rate of change per year: \[ \text{Average rate of change} = \frac{-4412}{3} \text{ million dollars per year} \] Simplified, this is approximately \[ -1470.67 \text{ million dollars per year} \]

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Key Concepts

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

Understanding Initial Value
The initial value is the starting point of the data you're examining. In this problem, the initial value represents the amount spent on analog TVs in 2003, which is 5836 million dollars. Knowing the initial value is essential as it sets the baseline for measuring any change over time. Think of it as the starting moment in your analysis journey.
Understanding Final Value
The final value marks the end of the period being analyzed. For our exercise, the final value is the spending on analog TVs in 2006, which is 1424 million dollars. Having both the initial and final values allows us to calculate how much change has occurred during the specific time frame under consideration.
The Importance of Time Period
The time period is the span over which you're measuring the change. In our exercise, it stretches from 2003 to 2006, totaling 3 years. Calculating the duration is crucial for determining the average rate of change, helping you understand the pace at which the changes occur.
Calculating Change in Sales
To find the change in sales, simply subtract the final value from the initial value. Here the calculation is: \[\text{Change in sales} = 1424 - 5836 = -4412 \text{ million dollars} \] The negative sign indicates a decrease. This shows that the amount spent on analog TVs significantly dropped during the THREE years. Understanding this change helps identify trends and make informed predictions about future sales.
Determining Average Rate of Change
To find the average rate of change, divide the change in sales by the number of years in the time period. For our problem, this is calculated as: \[\text{Average rate of change} = \frac{-4412}{3} \text{ million dollars per year} \] Simplified, this is approximately \[ -1470.67 \text{ million dollars per year} \] This negative average rate tells us that, on average, spending on analog TVs decreased by about 1470.67 million dollars each year over the given period. It's a useful measure to understand how quickly the sales are changing annually.

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