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In 2010, a local construction company built a total of 52 homes for a total cost of [tex]$\$[/tex]4.3[tex]$ million. By the end of 2010, they sold all 52 homes for a revenue of $[/tex]\[tex]$8$[/tex] million, thus making a profit of [tex]$\$[/tex]3.7[tex]$ million.

Since the company made a profit of $[/tex]\[tex]$3.7$[/tex] million, their net profit margin was
[tex]\[ \frac{\text{Profit}}{\text{Revenue}} = \frac{3.7}{8} = 46.25\% \][/tex]
which indicates they made [tex]$\$[/tex]0.4625[tex]$ for each dollar of revenue.

Since 2010, the cost of building materials and labor has steadily increased at a rate of $[/tex]\[tex]$0.31$[/tex] million per year. The selling price of housing has also increased but at a slower rate of [tex]$\$[/tex]0.08[tex]$ million per year.

Assume the cost of building homes and the revenue generated by selling the homes can be modeled by the following linear equations:

Cost in year $[/tex]x[tex]$: \[ C(x) = 0.31x + 4.3 \] (millions of dollars)

Revenue in year $[/tex]x[tex]$: \[ R(x) = 0.08x + 8 \] (millions of dollars)

Note: The variable $[/tex]x$ represents the number of years since 2010.



Answer :

GinnyAnswer
Sure, let's break down the problem step by step.

We are given that:
- The cost to build homes in year [tex]\( x \)[/tex] (where [tex]\( x \)[/tex] is the number of years since 2010) is given by the linear equation:
[tex]\[ C(x) = 0.31x + 4.3 \text{ (in millions of dollars) } \][/tex]

- The revenue from selling homes in year [tex]\( x \)[/tex] is given by the linear equation:
[tex]\[ R(x) = 0.08x + 8 \text{ (in millions of dollars) } \][/tex]

Let's calculate these values for the year 2020, which is 10 years after 2010 ([tex]\( x = 10 \)[/tex]):

1. Calculate the cost in 2020:
[tex]\[ C(10) = 0.31 \times 10 + 4.3 = 3.1 + 4.3 = 7.4 \text{ (in millions of dollars) } \][/tex]

2. Calculate the revenue in 2020:
[tex]\[ R(10) = 0.08 \times 10 + 8 = 0.8 + 8 = 8.8 \text{ (in millions of dollars) } \][/tex]

3. Calculate the profit in 2020:
Profit is defined as the revenue minus the cost.
[tex]\[ \text{Profit} = R(10) - C(10) = 8.8 - 7.4 = 1.4 \text{ (in millions of dollars) } \][/tex]

4. Calculate the net profit margin in 2020:
The net profit margin is calculated as:
[tex]\[ \text{Net Profit Margin} = \left( \frac{\text{Profit}}{\text{Revenue}} \right) \times 100\% \][/tex]
Substitute the profit and revenue values:
[tex]\[ \text{Net Profit Margin} = \left( \frac{1.4}{8.8} \right) \times 100 = 15.9090909090909 \% \][/tex]

To summarize, for the year 2020:
- The cost to build homes is \[tex]$7.4 million. - The revenue from selling homes is \$[/tex]8.8 million.
- The profit is \$1.4 million.
- The net profit margin is approximately [tex]\( 15.91\% \)[/tex].

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