Question

The way entertainment is sold and viewed has changed drastically in the past decade. In the year

2011, there were 6.1 billion DVDs sold worldwide. In the year 2021, the number had declined to
1.2 billion.

a. What is the average rate of change in DVD sales during this period?


b. Using the numbers in 2011 as an initial value, write a linear model that models the
number of DVD sales worldwide. Make sure to define the variables.


c. If the rate stays the same, predict the number of DVD sales in the
year 2023.

Answer 1

The average rate of change in DVD sales from 2011 to 2021 is -0.49 billion DVDs per year. A linear model representing this change would be D(t) = -0.49t + 6.1, with t being the years since 2011. Using this model, the predicted sales for 2023 would be 0.22 billion DVDs.

Step-by-step explanation:

The student is asking for assistance with calculating the average rate of change in DVD sales from 2011 to 2021, creating a linear model to represent this change, and using it to predict sales for the year 2023.

Part a: Average Rate of Change

The average rate of change is calculated by taking the difference in the sales figures and dividing by the time period over which that change occurred. In this case, we have:

Initial sales in 2011: 6.1 billion

Final sales in 2021: 1.2 billion

Change in sales: 6.1 - 1.2 = 4.9 billion

Time period: 2021 - 2011 = 10 years

The average rate of change = (4.9 billion) / (10 years) = 0.49 billion DVDs per year.

Part b: Linear Model

Let's denote D(t) as the number of DVD sales, where t represents the number of years since 2011. The linear model would then be:

D(t) = -0.49t + 6.1 (with t in years and D(t) in billions of DVDs)

Part c: Prediction for 2023

To predict the number of DVD sales in 2023, we set t = 2023 - 2011 = 12 years. Plugging this into our model, we get:

D(12) = -0.49(12) + 6.1 = -5.88 + 6.1 = 0.22 billion DVDs

Therefore, if the rate stays the same, the predicted number of DVD sales in the year 2023 is 0.22 billion.

Answer 2

Step-by-step explanation:

a. The average rate of change in DVD sales during this period can be calculated as:

average rate of change = (change in DVDs sold) / (change in years)

We are given that in 2011, 6.1 billion DVDs were sold, and in 2021, 1.2 billion DVDs were sold. The change in DVDs sold is:

change in DVDs sold = 1.2 billion - 6.1 billion = -4.9 billion

The change in years is:

change in years = 2021 - 2011 = 10

So the average rate of change in DVD sales during this period is:

average rate of change = (-4.9 billion) / 10 = -0.49 billion DVDs per year

Therefore, DVD sales declined by an average of 0.49 billion per year during this period.

b. Using the numbers in 2011 as an initial value, we can write a linear model that models the number of DVD sales worldwide as:

DVD sales = -0.49(t - 2011) + 6.1

where t is the year and DVD sales are measured in billions.

c. If the rate stays the same, we can predict the number of DVD sales in the year 2023 by substituting t = 2023 into the linear model and solving for DVD sales:

DVD sales = -0.49(2023 - 2011) + 6.1 ≈ 0.5 billion DVDs

Therefore, if the rate of decline in DVD sales stays the same, we can predict that there will be about 0.5 billion DVDs sold worldwide in 2023.