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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 profit.

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:
\[
\text{Cost in year } x: \ C(x)=0.31x+4.3 \ \text{millions of dollars}
\]
\[
\text{Revenue in year } x: \ R(x)=0.08x+8 \ \text{millions of dollars}
\]
Note the variable $[/tex]x[tex]$ represents the number of years since 2010.

\ \textless \ strong\ \textgreater \ Questions:\ \textless \ /strong\ \textgreater \

1. Calculate the company's net profit margin for 2018. Round all calculations to two decimal places.
a. Determine the value of $[/tex]x[tex]$ for 2018.
b. Determine the cost to build homes in 2018.
c. Determine the revenue for selling the homes in 2018.
d. Determine the profit made in 2018.
e. Determine the company's net profit margin for 2018. Give your answer rounded to four decimal places.

2. Calculate the company's net profit margin for 2022. Round all calculations to two decimal places.
a. Determine the value of $[/tex]x$ for 2022.
b. Determine the cost to build homes in 2022.
c. Determine the revenue for selling the homes in 2022.



Answer :

GinnyAnswer
Let's go through each question step-by-step.

### 1. Calculate the company's net profit margin for 2018:

a. Determine the value of [tex]\( x \)[/tex] for 2018.

2018 is 8 years after 2010, so:
[tex]\[ x = 2018 - 2010 = 8 \][/tex]

b. Determine the cost to build homes in 2018.

Using the cost equation [tex]\( C(x) = 0.31x + 4.3 \)[/tex]:
[tex]\[ C(2018) = 0.31 \cdot 8 + 4.3 = 6.78 \text{ million dollars} \][/tex]

c. Determine the revenue for selling the homes in 2018.

Using the revenue equation [tex]\( R(x) = 0.08x + 8 \)[/tex]:
[tex]\[ R(2018) = 0.08 \cdot 8 + 8 = 8.64 \text{ million dollars} \][/tex]

d. Determine the profit made in 2018.

Profit is revenue minus cost:
[tex]\[ \text{Profit} = 8.64 - 6.78 = 1.86 \text{ million dollars} \][/tex]

e. Determine the company's net profit margin for 2018.

Net profit margin is given by:
[tex]\[ \text{Net profit margin} = \frac{\text{Profit}}{\text{Revenue}} = \frac{1.86}{8.64} = 0.2153 \][/tex]
Rounded to four decimal places: [tex]\( 0.2153 \)[/tex] or [tex]\( 21.53\% \)[/tex].

### 2. Calculate the company's net profit margin for 2022:

a. Determine the value of [tex]\( x \)[/tex] for 2022.

2022 is 12 years after 2010, so:
[tex]\[ x = 2022 - 2010 = 12 \][/tex]

b. Determine the cost to build homes in 2022.

Using the cost equation [tex]\( C(x) = 0.31x + 4.3 \)[/tex]:
[tex]\[ C(2022) = 0.31 \cdot 12 + 4.3 = 8.02 \text{ million dollars} \][/tex]

c. Determine the revenue for selling the homes in 2022.

Using the revenue equation [tex]\( R(x) = 0.08x + 8 \)[/tex]:
[tex]\[ R(2022) = 0.08 \cdot 12 + 8 = 8.96 \text{ million dollars} \][/tex]

d. Determine the profit made in 2022.

Profit is revenue minus cost:
[tex]\[ \text{Profit} = 8.96 - 8.02 = 0.94 \text{ million dollars} \][/tex]

e. Determine the company's net profit margin for 2022.

Net profit margin is given by:
[tex]\[ \text{Net profit margin} = \frac{\text{Profit}}{\text{Revenue}} = \frac{0.94}{8.96} = 0.1049 \][/tex]
Rounded to four decimal places: [tex]\( 0.1049 \)[/tex] or [tex]\( 10.49\% \)[/tex].

In summary:
1. 2018 Calculations:
a. [tex]\( x = 8 \)[/tex]
b. Cost = [tex]\( \$ 6.78 \)[/tex] million
c. Revenue = [tex]\( \$ 8.64 \)[/tex] million
d. Profit = [tex]\( \$ 1.86 \)[/tex] million
e. Net Profit Margin = [tex]\( 0.2153 \)[/tex] or [tex]\( 21.53\% \)[/tex]

2. 2022 Calculations:
a. [tex]\( x = 12 \)[/tex]
b. Cost = [tex]\( \$ 8.02 \)[/tex] million
c. Revenue = [tex]\( \$ 8.96 \)[/tex] million
d. Profit = [tex]\( \$ 0.94 \)[/tex] million
e. Net Profit Margin = [tex]\( 0.1049 \)[/tex] or [tex]\( 10.49\% \)[/tex]

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