A company plans to sell a new type of vacuum cleaner for $280 each. The company's financial planner estimates that the cost, $y$, of manufacturing the vacuum cleaners is a quadratic function with a $y$-intercept of 11,000 and a vertex of (500, 24,000). Which system of equations can be used to determine how many vacuums must be sold for the company to make a profit?
A. [tex]\left\{\begin{array}{l}y=280 x \\ y=-0,052(x-500)^2+24,000\end{array}\right.[/tex]
B. [tex]\left\{\begin{array}{l}y=280 x \\ y=0.052 x^2+11,000\end{array}\right.[/tex]
C. [tex]\left\{\begin{array}{l}y=280 x \\ y=0.052(x-500)^2+24,000\end{array}\right.[/tex]
D. [tex]\left\{\begin{array}{l}y=280 x \\ y=-0.052(x-500)^2+11,000\end{array}\right.[/tex]
Answer (1)
The revenue equation is y = 280 x .
The cost function is a quadratic with vertex (500, 24000) and y-intercept 11000, leading to the equation y = a ( x − 500 ) 2 + 24000 .
Solving for 'a' using the y-intercept gives a = − 0.052 .
The system of equations is { y = 280 x y = − 0.052 ( x − 500 ) 2 + 24000 .
Explanation
Problem Analysis The problem asks for a system of equations that models the company's profit. The revenue is given by the price per vacuum cleaner times the number of vacuum cleaners sold, which is 280 x . The cost function is a quadratic with a vertex at (500, 24000) and a y-intercept of 11000. We need to find the correct quadratic equation representing the cost.
Revenue Equation The revenue equation is straightforward: y = 280 x . This represents the total income from selling x vacuum cleaners at $280 each.
Cost Function - Vertex Form The cost function is a quadratic, and we're given its vertex (500, 24000). The vertex form of a quadratic equation is y = a ( x − h ) 2 + k , where ( h , k ) is the vertex. So, we have y = a ( x − 500 ) 2 + 24000 .
Using the Y-Intercept We also know the y-intercept is 11000. This means when x = 0 , y = 11000 . We can use this information to find the value of a . Substituting these values into the equation, we get: 11000 = a ( 0 − 500 ) 2 + 24000 .
Solving for a Now, we solve for a : 11000 = a ( − 500 ) 2 + 24000
11000 = 250000 a + 24000
250000 a = 11000 − 24000
250000 a = − 13000
a = 250000 − 13000 = − 0.052
Cost Function - Complete Equation So, the cost function is y = − 0.052 ( x − 500 ) 2 + 24000 .
Final System of Equations Therefore, the system of equations is: { y = 280 x y = − 0.052 ( x − 500 ) 2 + 24000
Conclusion The system of equations that can be used to determine how many vacuums must be sold for the company to make a profit is: { y = 280 x y = − 0.052 ( x − 500 ) 2 + 24000
Examples
Understanding break-even points is crucial in business. For example, a bakery can use this system to determine how many cakes they need to sell to cover their costs, considering the price per cake and the costs of ingredients, labor, and rent. By setting up a revenue equation and a cost equation, they can find the point where revenue equals cost, indicating the number of cakes needed to be sold to start making a profit. This helps in making informed decisions about pricing, production levels, and overall business strategy.
