U $3$ D $8$ - Independent Practice $\newline$ $1$ . A small company that manufactures snowboards uses the relation $P=81 x(2-x)$ to model heir profit. In the model, $x$ represents the number of snowboards in thousands, and $P$ epresents the profit in thousands of dollars. $\newline$ a. The company breaks even where there is neither a profit or a loss. What are the break even points for the company? $\newline$ b. How many snowboards must the company produce to earn a maximum profit? $\newline$ c. What is the maximum profit the company can earn?

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Grade 8

Interpreting Linear Expressions

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Q. U

3

8

- Independent Practice

\newline

1

. A small company that manufactures snowboards uses the relation

P=81 x(2-x)

to model heir profit. In the model,

x

represents the number of snowboards in thousands, and

P

epresents the profit in thousands of dollars.

\newline

a. The company breaks even where there is neither a profit or a loss. What are the break even points for the company?

\newline

b. How many snowboards must the company produce to earn a maximum profit?

\newline

c. What is the maximum profit the company can earn?

Set $P$ equal to $0$ : To find the break even points, set $P$ equal to $0$ and solve for $x$ . $0 = 81 \times (2 - x)$
Isolate the term: Divide both sides by $81$ to isolate the term $(2 - x)$ . $\newline$ $0 = 2 - x$
Solve for x: Add $x$ to both sides to solve for $x$ . $x = 2$
Break even point: The break even point is when $x$ equals $2$ , which means the company breaks even when they sell $2000$ snowboards.
Find the vertex: To find the number of snowboards for maximum profit, we need to find the vertex of the parabola represented by the equation $P = 81 \times (2 - x)$ . The vertex form of a parabola is $P = a(x - h)^2 + k$ , where $(h, k)$ is the vertex.
Rewrite the equation: The equation $P = 81 \times (2 - x)$ can be rewritten as $P = -81x^2 + 162x$ by expanding and rearranging.
Calculate h: The $x$ -coordinate of the vertex ( $h$ ) can be found using the formula $h = -\frac{b}{2a}$ . $\newline$ In our equation, $a = -81$ and $b = 162$ .
Determine snowboards needed: Calculate $h = -162 / (2 \times -81)$ . $h = -162 / -162$
Substitute $x$ into equation: $h = 1$ $\newline$ This means the company must produce $1000$ snowboards (since $x$ is in thousands) to earn maximum profit.
Calculate maximum profit: To find the maximum profit $k$ , substitute $x = 1$ into the equation $P = -81x^2 + 162x$ . $P = -81(1)^2 + 162(1)$
Calculate maximum profit: To find the maximum profit $k$ , substitute $x = 1$ into the equation $P = -81x^2 + 162x$ . $P = -81(1)^2 + 162(1)$ Calculate $P = -81 + 162$ . $P = 81$
Calculate maximum profit: To find the maximum profit $k$ , substitute $x = 1$ into the equation $P = -81x^2 + 162x$ . $P = -81(1)^2 + 162(1)$ Calculate $P = -81 + 162$ . $P = 81$ The maximum profit the company can earn is $\$81,000$ (since $P$ is in thousands of dollars).