Problem 104 - Calculus Volume 1

Question

Total online shopping during the Christmas holidays has increased dramatically during the past 5 years. In \(2012 \quad(t=0),\) total online holiday sales were $$\$ 42.3$$ billion, whereas in 2013 they were \(\$ 48.1\) billion. a. Find a linear function S that estimates the total online holiday sales in the year t . b. Interpret the slope of the graph of S. c. Use part a. to predict the year when online shopping during Christmas will reach 60 billion.

Step-by-Step Solution

Answer

The sales will reach \$60 billion in 2015.

Step 1: Identify Known Values

From the problem, we identify that at \(t = 0\), which corresponds to the year 2012, the sales were \\(42.3\ billion. In 2013, which is \(t = 1\), the sales were \\)48.1\ billion. We have two points: \((0, 42.3)\) and \((1, 48.1)\). These points will be used to form the linear equation.

Step 2: Calculate the Slope (m)

The slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Using our points \((0, 42.3)\) and \((1, 48.1)\), we can calculate:\[m = \frac{48.1 - 42.3}{1 - 0} = \frac{5.8}{1} = 5.8\]

Step 3: Write the Linear Equation

The linear function \(S(t)\) can be written in slope-intercept form as \(S(t) = mt + b\). We know \(m = 5.8\) and using the point \((0, 42.3)\), the y-intercept \(b\) is 42.3. Thus, the equation becomes:\[S(t) = 5.8t + 42.3\]

Step 4: Interpret the Slope

The slope \(5.8\) represents the rate of increase in online holiday sales per year. Each year, online holiday sales increase by \$5.8 billion.

Step 5: Solve for Year when Sales Reach 60 Billion

We are asked to find the year when sales \(S(t)\) reach \$60 billion. We set up the equation:\[ 5.8t + 42.3 = 60\]Subtract 42.3 from both sides:\[ 5.8t = 17.7\]Divide both sides by 5.8:\[ t = \frac{17.7}{5.8} \approx 3.05\]Since \(t\) represents the number of years after 2012, we round to obtain the estimated year. Thus, \(t \approx 3\) and the year would be 2015.

Key Concepts

Slope Interpretation

When interpreting the slope in the context of a linear function, you're essentially trying to understand how one variable changes in relation to another. For a linear equation, the slope tells you how much the dependent variable, in this case, the total online holiday sales, increases for each one-unit increase in the independent variable, which here is time measured in years.
In the given exercise, the slope calculated is 5.8. This value has a significant meaning:

• It indicates that every year, starting from 2012, online holiday sales increased by $5.8 billion.
• This consistent yearly increase helps us analyze past trends and predict future sales.
• An important aspect here, particularly for stakeholders, is understanding the potential market growth over the short and long term based on these increments.

Understanding the slope provides valuable insights into market dynamics and helps in planning subsequent marketing and operational strategies.

Linear Equation Formation

Forming a linear equation from data involves finding a relationship expressed clearly in the formula of a line: the slope-intercept form: \(y = mx + b\). Here, `\(m\)` is the slope or rate of change, and `\(b\)` is the y-intercept, the starting point of the variable value when calculations begin at \(x = 0\).
In our exercise, using the data points (0, 42.3) and (1, 48.1), we've derived the linear equation:

• First, calculate the slope \(m\) using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
• For this example, substitute into the calculation: \(m = \frac{48.1 - 42.3}{1 - 0} = 5.8\).
• Next, utilize the point \((x, y) = (0, 42.3)\) to determine the y-intercept, b, which is 42.3, since when \(x\) equals 0, \(y\) equals \(b\).
• Thus, the equation becomes \(S(t) = 5.8t + 42.3\).

This linear equation allows us to make predictions, as evidenced by its use in predicting when sales reach $60 billion, thus illustrating the power of linear models in real-world applications.

Data Analysis in Mathematics

Data analysis in mathematics often employs linear functions for simpler prediction and trend analysis. By aligning data into a linear relationship, we can make educated guesses about the future.
Consider the sales data given in the exercise, which shows a consistent upward trend. Using a linear function:

• We can predict the year online holiday sales will hit $60 billion.
• By solving the equation \(5.8t + 42.3 = 60\), we determine that this milestone is reached approximately 3 years after 2012, so around 2015.
• This approach showcases how linear functions convert actual data into a model that forecasts future possibilities.

Data analysis with mathematics isn't just about number-crunching; it's about harnessing those numbers to project what may come next. This utility makes linear functions a staple in domains extending from economics to technology and beyond.