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Zoe paints landscapes and sells them at the Archbury Annual Art Market. She has discovered that the more she charges for each painting, the fewer of her paintings she is able to sell. Her total revenue from selling the paintings can be modeled by the expression p(960−4p)p(960 - 4p), where pp is the price per painting in dollars. This expression can be written in factored form as −4p(p−240)-4p(p - 240).\newlineWhat does the number 240240 represent in the expression?\newline(A)the price per painting in dollars so that Zoe's total revenue is zero\newline(B)the price per painting in dollars that maximizes Zoe's total revenue\newline(C)Zoe's minimum total revenue in dollars\newline(D)Zoe's maximum total revenue in dollars

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Q. Zoe paints landscapes and sells them at the Archbury Annual Art Market. She has discovered that the more she charges for each painting, the fewer of her paintings she is able to sell. Her total revenue from selling the paintings can be modeled by the expression p(960−4p)p(960 - 4p), where pp is the price per painting in dollars. This expression can be written in factored form as −4p(p−240)-4p(p - 240).\newlineWhat does the number 240240 represent in the expression?\newline(A)the price per painting in dollars so that Zoe's total revenue is zero\newline(B)the price per painting in dollars that maximizes Zoe's total revenue\newline(C)Zoe's minimum total revenue in dollars\newline(D)Zoe's maximum total revenue in dollars
  1. Revenue Expression Analysis: Zoe's total revenue is given by the expression p(960−4p)p(960 - 4p). To find what 240240 represents, we need to understand the factored form −4p(p−240)-4p(p - 240).
  2. Factored Form Interpretation: The factored form −4p(p−240)-4p(p - 240) shows that the revenue is zero when p=0p = 0 or p=240p = 240. This means that if Zoe charges $240\$240 per painting, she will sell zero paintings, making her revenue $0\$0.
  3. Price and Revenue Relationship: However, the question asks what 240240 represents in terms of Zoe's revenue. Since the revenue is zero at p=240p = 240, this is the price at which Zoe's revenue starts to decrease if she charges more.
  4. Optimal Price for Revenue Maximization: Therefore, $\(240\) is the price per painting in dollars that maximizes Zoe's total revenue. If she charges more than \(\$240\), she will sell fewer paintings, and her revenue will decrease.

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