91Ó°ÊÓ

Two companies sell software products. In 2010 , Company 1 had total sales of \(\$ 17.2\) million. Its marketing department projects that sales will increase by \(\$ 1.5\) million per year for the next several years. Company 2 had total sales of \(\$ 9.6\) million for software products in 2010 and predicts that its sales will increase \(\$ 2.3\) million each year on average. Let \(x\) represent the number of years since 2010 . a. Write an equation that represents the total sales, in millions of dollars, of Company 1 since 2010 . Let \(S\) represent the total sales in millions of dollars. b. Write an equation that represents the total sales, in millions of dollars, of Company 2 since 2010 . Let \(S\) represent the total sales in millions of dollars. c. Write a single equation to determine when the total sales of the two companies will be the same. d. Solve the equation in part \(c\) and interpret the result.

Step by step solution

01

Write the equation for total sales of Company 1

Company 1 has an initial sales of \(17.2\) million in 2010, and it's increasing by \(1.5\) million each year. So, its total sales can be represented as a linear function: \(S_1 = 17.2 + 1.5x\), where \(S_1\) is the total sales of Company 1 in millions of dollars and \(x\) represents the number of years since 2010.

02

Write the equation for total sales of Company 2

Company 2 has an initial sales of \(9.6\) million in 2010, and it's increasing by \(2.3\) million each year. So, its total sales can be represented as a linear function: \(S_2 = 9.6 + 2.3x\), where \(S_2\) is the total sales of Company 2 in millions of dollars and \(x\) represents the number of years since 2010.

03

Write an equation representing when the sales of both companies will be equal

Let's find when the total sales of the two companies will be the same, i.e., \(S_1 = S_2\). We can set up the equation as follows: $$17.2 + 1.5x = 9.6 + 2.3x$$

04

Solve the equation and interpret the result

Now, we need to solve the equation for \(x\) to find out when the total sales of both companies will be the same. \begin{align*} 17.2 + 1.5x &= 9.6 + 2.3x \\ 1.5x - 2.3x &= 9.6 -17.2 \\ -0.8x &= -7.6 \\ x &= \frac{-7.6}{-0.8} \\ x &= 9.5 \end{align*} The result is \(x=9.5\), which means that the total sales of both companies will be equal after about \(9.5\) years since 2010. We could interpret that the sales of both companies will be the same around mid-year of 2020 (since 2010 + 9.5 = 2019.5).

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