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Chapter 4: Problem 86

A model of the amount of e-commerce revenue for rental and leasing services companies, \(y\), in \(x\) years after 2005 is \(y=\left(\frac{\$ 8.71 \text { billion }}{1 \text { year }}\right) x+\$ 6.01\) billion. Use the model to find the amount of e-commerce revenue in \(2012 .\) (Source: www.census.gov, 2011)

Short Answer

Expert verified
Approximately \$66.98 billion.

Step by step solution

01

Identify the given model

The model provided is: \[ y = \left(\frac{8.71 \text{ billion}}{1 \text{ year}}\right) x + 6.01 \text{ billion} \]. This equation relates the e-commerce revenue, \(y\), to years after 2005, \(x\).
02

Determine the value of \(x\)

To find the revenue in 2012, calculate the number of years after 2005: \[ x = 2012 - 2005 = 7 \text{ years} \].
03

Substitute \(x\) into the model

Replace \(x\) with 7 in the equation: \[ y = 8.71 \cdot 7 + 6.01 \].
04

Calculate the result

First multiply 8.71 by 7: \[ 8.71 \times 7 = 60.97 \]. Then add 6.01: \[ 60.97 + 6.01 = 66.98 \text{ billion} \].

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Key Concepts

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

linear equations
Linear equations are fundamental in algebra. They involve variables whose highest power is one, and the graph of a linear equation is a straight line. The general form is usually written as: \[ y = mx + b \]. Here, \(m\) represents the slope, and \(b\) the y-intercept. In our exercise, we saw a linear equation representing the e-commerce revenue over time. Understanding linear equations helps in predicting trends and solving numerous real-world problems.
substitution method
The substitution method is a key technique for solving equations, especially in systems of linear equations. The idea is to substitute one equation into another to find the value of the variables involved. Let's break it down:
  • Start with one equation and solve for one variable in terms of the other.
  • Next, substitute this variable back into the other equation.
  • Solve the new equation for the remaining variable.
In our example, we substituted \(x = 7\) into the linear model, enabling us to find the e-commerce revenue for 2012.
real-life applications of algebra
Algebra is not just an abstract mathematical field. It finds real-life applications everywhere. Take the given exercise: the model \( y = 8.71x + 6.01 \) helps to predict e-commerce revenue over years. This model can assist businesses in planning and decision-making.
Aside from e-commerce:
  • Finance uses algebra for budgeting and projecting profits.
  • Engineering relies on it for designing structures and processes.
  • Medicine uses algebra for pharmacokinetics and disease modeling.
Thus, mastering algebra opens up doors to a multitude of practical, impactful opportunities.

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Most popular questions from this chapter

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