A company has determined that its weekly profit is a function of the number of items that it sells. Which equation could represent the weekly profit in thousands of dollars, [tex]y[/tex], when the company sells [tex]x[/tex] items?
A. [tex]y^2 = 4x^2 - 100[/tex] B. [tex]y = -x^2 + 50x - 300[/tex] C. [tex]x = -y^2 + 60y - 400[/tex] D. [tex]x^2 = -6y^2 + 200[/tex]
To determine which equation could represent the weekly profit in thousands of dollars, [tex]\(y\)[/tex], as a function of the number of items [tex]\(x\)[/tex] that the company sells, we need to analyze the given equations with respect to their suitability in depicting such a relationship:
1. Equation [tex]\(y^2 = 4x^2 - 100\)[/tex]: - This equation implies that [tex]\( y \)[/tex] (weekly profit) squared is a function of [tex]\( x \)[/tex] (number of items) squared with a constant subtracted. - While it is mathematically valid, it might not practically model typical profit behaviors straightforwardly.
2. Equation [tex]\(y = -x^2 + 50x - 300\)[/tex]: - This is a quadratic equation in the form of [tex]\( y = ax^2 + bx + c \)[/tex]. - Quadratic equations are commonly used to model profit scenarios because they can depict an increase in profit up to a certain point (maximum profit) and then a decrease, which reflects real-world situations where profits increase when sales start but might decrease after reaching a saturation point or due to increasing costs. - This equation logically fits a realistic scenario for weekly profit as it can describe the behavior where profit initially increases with sales and then eventually decreases after a certain point.
3. Equation [tex]\(x = -y^2 + 60y - 400\)[/tex]: - This equation solves for [tex]\( x \)[/tex], making [tex]\( x \)[/tex] a function of [tex]\( y \)[/tex]. - In the given context, [tex]\(y\)[/tex] should be the dependent variable (profit) based on the number of items [tex]\( x \)[/tex] sold, which doesn't align well with this equation's structure where [tex]\( x \)[/tex] depends on [tex]\( y \)[/tex].
4. Equation [tex]\(x^2 = -6y^2 + 200\)[/tex]: - This equation suggests a relationship where [tex]\( x^2 \)[/tex] is a function of [tex]\( y^2 \)[/tex], making it a rather complex representation and not intuitively fitting a profit-sales relationship. - It seems quite unusual and might not logically represent the typical behavior between the weekly profit and the number of items sold.
After evaluating all these equations, the most appropriate one that fits the logical and practical scenario of a company's weekly profit being a function of the number of items sold is:
[tex]\[ y = -x^2 + 50x - 300 \][/tex]
Thus, the correct answer is the second equation: [tex]\[ y = -x^2 + 50x - 300 \][/tex]
