Question

in the first equation in the system of equations, y represents the money collected from selling sweatshirts. in the second equation, y represents the money spent to produce x sweatshirts with team logos on them for a professional sports league. what does the solution of the system represent in this context?

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$$\begin{cases} y = 35x \\\\ y = -0.05(x - 400)^2 + 9,492 \\end{cases}$$

\bigcirc the number of sweatshirts that generate the maximum income
\bigcirc the number of sweatshirts that generate the minimum cost
\bigcirc the number of sweatshirts for which the difference between cost and income is greatest
\bigcirc the number of sweatshirts for which cost and income are equal

Explanation:

In a system of equations, the solution is the point where the two equations are equal (i.e., where their graphs intersect). Here, the first equation \( y = 35x \) represents income (money from selling \( x \) sweatshirts), and the second equation \( y=-0.05(x - 400)^2+9492 \) represents cost (money spent to produce \( x \) sweatshirts). So, solving the system means finding \( x \) (and \( y \)) where income (\( y \) from first equation) equals cost (\( y \) from second equation).

• Option 1: The maximum income is from the vertex of the quadratic (since it's a downward - opening parabola), not the system's solution.
• Option 2: The minimum cost isn't related to the intersection of these two equations.
• Option 3: The greatest difference between cost and income isn't found at the intersection (intersection is where they are equal, difference is zero there).
• Option 4: The solution of the system gives the \( x \) (number of sweatshirts) where income (from selling) and cost (from production) are equal.

Answer:

D. the number of sweatshirts for which cost and income are equal