Problem 61 - Applied Calculus

Question

Recently, tablet computer sales in the United States have been growing approximately linearly. In 2011 sales were 70 million units, and in 2013 sales were 146 million units. a. Use the two (year, sales) data points (1,70) and (3,146) to find the linear relationship \(y=m x+b\) between \(x=\) years since 2010 and \(y=\) sales (in millions). b. Interpret the slope of the line. c. Use the linear relationship to predict sales in the year \(2020 .\) Note: Tablet computers include iPads, Kindles, and Nooks.

Step-by-Step Solution

Answer

Sales increase by 38 million units per year; projected sales in 2020 are 412 million units.

Step 1: Identify the Two Points

In this problem, we are given two data points: (1, 70) for the year 2011 and (3, 146) for the year 2013. These represent the sales in millions in year format where 2010 corresponds to x = 0.

Step 2: Calculate the Slope (m)

The formula for the slope of a line is given by \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Substitute the given points (1, 70) and (3, 146):\[ m = \frac{146 - 70}{3 - 1} = \frac{76}{2} = 38 \]Thus, the slope is 38.

Step 3: Find the y-intercept (b)

Using the slope-intercept form of the equation, \( y = mx + b \), and substituting one of the points (1, 70), we find \( b \):\[ 70 = 38 \times 1 + b \]\[ b = 70 - 38 = 32 \]So, the y-intercept is 32.

Step 4: Write the Linear Equation

Now that we have both the slope and the y-intercept, the linear equation can be written as:\[ y = 38x + 32 \]

Step 5: Interpret the Slope

The slope of the line, 38, implies that each year, the sales of tablet computers increase by an average of 38 million units.

Step 6: Predict Sales in 2020

To predict sales for the year 2020, we need to determine the value of \( x \) in 2020, which is 10 (since \( x = 2020 - 2010 \)). Substitute \( x = 10 \) into the equation:\[ y = 38 \times 10 + 32 = 380 + 32 = 412 \]The predicted sales in 2020 are 412 million units.

Key Concepts

Slope Calculation

Understanding how to calculate the slope is crucial in analyzing linear relationships between two variables. The slope represents the rate of change. In our exercise, the points given are (1, 70) and (3, 146), where the first number denotes the year since 2010, and the second represents sales in millions.
To find the slope (denoted as \(m\)), you use the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

Inserting our data, we get:

• \( m = \frac{146 - 70}{3 - 1} = \frac{76}{2} = 38 \)

This calculation shows that each additional year corresponds to an increase of 38 million units in tablet sales. The slope tells us how drastically sales numbers rise over time.
This concept is vital for predicting future trends or evaluating past performances in similar linear patterns.

Linear Equation

A linear equation provides a straightforward way to depict relationships in data. The standard form we use is \( y = mx + b \), where \( m \) is the slope, and \( b \) is the intercept on the y-axis.
In this exercise, we already discovered the slope to be 38. To find the y-intercept \(b\), which represents the initial value before any increase, we use one of our data points, say (1, 70):

• \( 70 = 38 \times 1 + b \)
• Solving for \( b \), we find: \( b = 70 - 38 = 32 \)

Therefore, the complete linear equation becomes:

• \( y = 38x + 32 \)

This equation is critical because it allows us to predict sales figures over time by just plugging in the value of \( x \), representing years since 2010.

Data Interpretation

Interpreting linear regression results allows us to make meaningful conclusions and predictions based on data. Here, the slope of 38 tells us that sales are expected to grow by 38 million units each year. This rate of increase gives a clear insight into the tablet market's expansion.
The linear formula \( y = 38x + 32 \) allows us to predict, with reasonable accuracy, the sales for any given year by substituting the value of \( x \).
For example, to forecast sales in 2020, we calculate \( x \) as 10 (since 2020 - 2010 = 10), and substitute into our equation:

• \( y = 38 \times 10 + 32 \)
• \( y = 380 + 32 = 412 \)

The prediction of 412 million units in 2020 highlights the power of interpreting and utilizing linear models for forecasting future outcomes based on current trends.